European Chapter on
Combinatorial Optimization
(ECCO) XXII
May 17-19, 2009
Jerusalem
Abstracts booklet
The conference is supported by:
Sunday
9:15-10:15 Plenary talk – Alain Hertz "Solution methods for the inventory routing
problem"
We consider an inventory routing problem in discrete time where a supplier has to
serve a set of customers over a time horizon. At each discrete time, a fixed quantity is
produced at the supplier and a fixed quantity is consumed at each customer. A
capacity constraint for the inventory is given for the supplier and each customer, and
the service cannot cause any stock-out situation. Two different replenishment policies
are considered: when a customer is served, one can either deliver any quantity that
does not violate the capacity constraint, or impose that the inventory level at the
customer should reach its maximal value. The transportation cost is proportional to
the distance traveled, whereas the inventory holding cost is proportional to the level of
the inventory at the customers and at the supplier. The objective is the minimization
of the sum of the inventory and transportation costs. We describe the current best
exact and heuristic algorithms for the solution of this complex problem.
10:45-12:00 Parallel sessions:
Track 1: Integer and mixed integer formulations 1
1. Julia Rieck and Jürgen Zimmermann, " A new mixed integer linear model for a
rich vehicle routing problem with docking constraints"
In this paper we address a rich vehicle routing problem that arises in real-life
applications. Among other aspects we consider
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heterogenous vehicles,
simultaneous delivery and pick-up at customer locations,
a total working time for vehicles from departure to arrival back at the depot,
time windows for customers during which delivery and pick-up can occur,
a time window for the depot, and
multiple use of vehicles throughout the planning horizon.
To guarantee a coordinated material flow at the depot, we include the timed allocation
of vehicles to loading bays at which the (un-)loading activities can occur. The
resulting rich vehicle routing problem can be formulated as a two-index vehicle-flow
formulation where we use binary variables to indicate if a vehicle traverses an arc in
the optimal solution and to sequence the (un-)loading activities at the depot.
In our performance analysis, we use CPLEX 11.0 to solve instances that are
derived from the extended Solomon test set. The selective implementation of
preprocessing techniques and cutting planes improves the solver performance
significantly. In this context, we strengthen the domains of auxiliary variables as well
as the big-M-constraints of our model. We identified clique, implied bound, and flow
cover cuts as particularly suitable for our purposes. Additionally, we take into
consideration the rounded capacity cuts for the capacitated vehicle routing problem.
As average computational time for instances with 25 customers we obtain 310.06
seconds and for instances with 30 customers 1305.10 seconds. A comparison to the
results obtained for a three-index vehicle-flow formulation shows that the model is
suitable for practical applications.
2. Bartosz Sawik, "Bi-objective dynamic portfolio optimization by mixed integer
programming"
This paper presents the dynamic portfolio optimization problem formulated as a biobjective mixed integer linear program. The computational efficiency of MILP model
is very important for applications to real-life financial and other decisions where the
constructed portfolios have to meet numerous side constraints. The portfolio selection
problem considered is based on a multi-period model of investment, in which the
investor buys and sells securities in consecutive periods. An extension of the
Markowitz portfolio optimization model is considered, in which the variance has been
replaced with the Value-at-Risk (VaR). The VaR is a quantile of the return
distribution function. The advantage of using this measure in portfolio optimization is
that this value of risk is independent of any distribution hypothesis.
The portfolio selection problem is usually considered as a bi-objective optimization
problem where a reasonable trade-off between expected rate of return and risk is
sought. The objective is to maximize future returns by picking the best amount of
stocks. The only way to improve future returns is to increase the risk level that the
decision maker is willing to accept. In the classical Markowitz approach, future
returns are random variables controlled by such parameters as the portfolio efficiency,
which is measured by the expectation, while risk is calculated by the standard
deviation. As a result the classical problem is formulated as a quadratic program with
continuous variables and some side constraints.
The objective of the problem considered in this paper is to allocate wealth on different
securities to maximize the portfolio expected return and the threshold of the
probability that the return is not less than a required level. The auxiliary objective is
minimization of risk probability of portfolio loss.
The input data for computations consist of 3500 historic daily quotations divided into
14 investment periods. The results of some computational experiments with the mixed
integer programming approach modeled on a real data from the Warsaw Stock
Exchange are reported.
3. Alberto Ceselli, Giovanni Righini and Gregorio Tirado Dominguez,
"Mathematical programming algorithms for the double TSP with multiple stacks"
The Double TSP with Multiple Stacks requires to find a minimum cost pair of
Hamiltonian tours, visiting two given sets of pickup and delivery customers, such that
the two sequences of visits can be executed by a single vehicle that can trasnport
goods organizing them in stacks of identical capacity. Stacks are managed according
to a LIFO policy and items cannot be rearranged. For this problem only heuristic
algorithms have been published so far.
We present a decomposition approach that leads to Lagrangian relaxation and column
generation algorithms. We also present some preliminary computational results.
Track 2: Approximation and Online algorithms:
1. Zvi Lotker, Boaz Patt-Shamir and Dror Rawitz, "Rent, lease or buy: randomized
strategies for multislope ski rental"
In the Multislope Ski Rental problem, the user needs a certain resource for some
unknown period of time. To use the resource, the user must subscribe to one of
several options, each of which consists of a one-time setup cost (“buying price”), and
cost proportional to the duration of the usage (“rental rate”). The larger the price, the
smaller the rent. The actual usage time is determined by an adversary, and the goal of
an algorithm is to minimize the cost by choosing the best option at any point in time.
Multislope Ski Rental is a natural generalization of the classical Ski Rental problem
(where the only options are pure rent and pure buy), which is one of the fundamental
problems of online computation. The Multislope Ski Rental problem is an abstraction
of many problems, where online choices cannot be modeled by just two alternatives,
e.g., power management in systems which can be shut down in parts.
In this work we study randomized online strategies for Multislope Ski Rental.
Our results include an algorithm that produces the best possible randomized online
strategy for any additive instance, where the cost of switching from one alternative to
another is the difference in their buying prices; and an e-competitive randomized
strategy for any (non-additive) instance. We also provide a randomized strategy with
a matching lower bound for the case of two slopes, where both slopes have positive
rents.
2. Asaf Levin and Uri Yovel, "Algorithms for (p,k)-uniform unweighted set cover
problem"
We are given n base elements and a finite collection of subsets of them. The size of
any subset varies between p to k (p<k). In addition, we assume that the input contains
all possible subsets of size p. Our problem is to find a subcollection of minimumcardinality which covers all the elements. This problem is known to be NP-hard. We
provide two approximation-algorithms for it, one for the generic case, and an
improved one for the special case of (p,k) = (2,4).
The algorithm for the generic case is a greedy one, based on packing phases: at each
phase we pick a collection of disjoint subsets covering i new elements, starting from
i=k down to i=p+1. At a final step we cover the remaining base elements by the
subsets of size p. We derive the exact approximation ratio of this algorithm for all
values of k and p, which is less than Hk, where Hk is the k’th harmonic number.
However, the algorithm exhibits the known improvement methods over the greedy
one for the unweighted k-set cover problem (in which subset sizes are only restricted
not to exceed k), and hence it serves as a benchmark for our improved algorithm.
The improved algorithm for the special case of (p,k) = (2,4) is based on local-search:
it starts with a feasible cover, and then repeatedly tries to replace sets of size 3 and 4
so as to maximize an objective function which prefers big sets over small ones. For
this case, our generic algorithm achieves an approximation ratio of 1.5, and the localsearch algorithm achieves a better ratio, which is bounded by 1.458333...
3. Leah Epstein and Gyorgy Dosa, "Preemptive online scheduling with reordering"
We consider online preemptive scheduling of jobs, arriving one by one, on m parallel
machines. A buffer of fixed size K>0, which assists in partial reordering of the input,
is available, to be used for the storage of at most K unscheduled jobs. We consider
several variants of the problem. For general inputs and identical machines, we show
that a buffer of size Theta(m) reduces the overall competitive ratio from e/(e-1) to 4/3.
Surprisingly, the competitive ratio as a function of m is not monotone, unlike the case
where K=0.
12:10-13:00 Parallel sessions:
Track 1: Integer and mixed integer formulations 2:
1. Jan Pelikán and Jakub Fischer, "Optimal doubling models"
The article deals with the problem of doubling the part of the system in order to
increase the reliability of this system. The task is to find out which part should be
doubled if it is not possible to double all of them for economic reasons. The goal is to
maximize the system’s reliability characterized by the probability of the faultless
functioning or to minimize the losses caused by the system’s failure. The article
proposes a procedure based on linear programming methods that find the optimal
decision, which of the parts should be doubled.
2. Jorge Riera-Ledesma and Juan José Salazar González, "A branch-and-cut-andprice for the multiple vehicle traveling purchaser problem"
The Multiple Vehicle Traveling Purchaser Problem aims designing a set of optimal
school bus routes with the purpose of carrying pupils to their center of learning. The
problem is defined as follows: given a set of users going to a specific location, let's
say school, there exist a set of potential bus stops. Each bus stop is reachable by a
subset of users. A fleet of homogeneous vehicles is available with this purpose. The
aim is assigning the set of users to a subset of potential stops, finding least cost trips
that starts and ends at the school, and choosing a vehicle to serve each trip so that
each stop with user assigned is served by a trip, and the total number of users assigned
to the stops of a trip does not exceed the capacity of the vehicle.
We propose an integer linear formulation, and several families of valid inequalities
are derived to strengthen the linear relaxation. A branch-and-cut-and-price procedure
has been developed, and a extensive computational experience is presented.
Track 2: Approximation algorithms for bin packing problems:
1. Rolf Harren and Rob van Stee, "An absolute 2-approximation algorithm for twodimensional bin packing"
We consider the problem of packing rectangles into an unlimited supply of equallysized, rectangular bins. All rectangles have to be packed non-overlapping and
orthogonal, i.e., axis-parallel. The goal is to minimize the number of bins used.
In general, multi-dimensional packing is strongly related to scheduling, as one
dimension can be associated with the processing time of the tasks (items) and the
other dimensions make up geometrically ordered resources. In two-dimensional
packing, e.g., tasks have to be processed on consecutive machines. The bin packing
setting that we consider here applies when the machines work in shifts, e.g., only
during the daytime. Thus the schedule has to be fixed into separate blocks.
Most of the previous work on two-dimensional bin packing has focused on the
asymptotic approximation ratio, i.e., allowing a constant number of spare bins. In this
setting, algorithms by Caprara and by Bansal et al. have approximation ratio better
than 2. On the other hand, the additive constants are (very) large and thus these
algorithms give bad approximation ratios for instances with small optimal value.
In terms of absolute approximability, i.e., without wasting additional bins, the
previously best-known algorithms have a ratio of 3. If rotations by 90 degrees are
permitted, the problem becomes somewhat easier as we do not have to combine
differently oriented thin items into a bin. We previously showed an absolute 2approximation for this case.
Here we present an algorithm for the problem without rotations with an absolute
worst-case ratio of 2, i.e., our algorithm uses at most twice the optimal number of
bins. It was shown that it is strongly NP-complete to decide wether a set of squares
can be packed into a given square. Therefore our result is best possible unless P = NP
and we thus settle the question of absolute approximability of two-dimensional bin
packing.
Our algorithm consists of three main parts depending on the optimal value of the
given instance. As we do not know this optimal value in advance, we apply all three
algorithms and allow them to fail if the requirement on the optimal value is not
satisfied. The algorithms with asymptotic approximation ratio less than 2 give an
absolute 2-approximation for instances with large optimal value. We design further
algorithms to solve instances that fit into one bin and that fit into a constant number of
bins, respectively. All three algorithms together make up an absolute 2approximation.
2. Leah Epstein and Asaf Levin, "Bin packing with general cost structures"
Following the work of Anily et al., we consider a variant of bin packing, called bin
packing with general cost structures and design an asymptotic fully polynomial time
approximation scheme (AFPTAS) for this problem. In the classic bin packing
problem, a set of one-dimensional items is to be assigned to subsets of total size at
most 1, that is, to be packed into unit sized bins. However, in our problem, the cost of
a bin is not 1 as in classic bin packing, but it is a non-decreasing and concave function
of the number of items packed in it, where the cost of an empty bin is zero. The
construction of the AFPTAS requires novel techniques for dealing with small items,
which are developed in this work. In addition, we develop a fast approximation
algorithm which acts identically for all non-decreasing and concave functions, and has
an asymptotic approximation ratio of 1.5 for all functions simultaneously.
14:00-15:15 Parallel sessions:
Track 1: Algorithms:
1. Igor Averbakh, "Minmax regret bottleneck problems with solution-induced
interval uncertainty structure"
We consider minmax regret bottleneck subset-type combinatorial optimization
problems, where feasible solutions are some subsets of a finite ground set. The
weights of elements of the ground set are uncertain; for each element, an uncertainty
interval that contains its weight is given. In contrast with previously studied interval
data minmax regret models, where the set of scenarios (possible realizations of the
vector of weights) does not depend on the chosen feasible solution, we consider the
problem with solution-induced interval uncertainty structure. That is, for each element
of the ground set, a nominal weight from the corresponding uncertainty interval is
fixed, and it is assumed that only the weights of the elements included in the chosen
feasible solution can deviate from their respective nominal values. We present a
number of algorithmic results for bottleneck minmax regret problems of this type, in
particular, a quadratic algorithm for the problem on a uniform matroid.
2. Endre Boros, Ondrej Cepek, Alex Kogan and Petr Kucera, "A subclass of Horn
CNFs optimally compressible in polynomial time"
The problem of Horn Minimization (HM) can be stated as follows: given a Horn CNF
representing a Boolean function f, find a CNF representation of f which consists of a
minimum possible number of clauses. This is a classical combinatorial optimization
problem with many practical applications. For instance, the problem of knowledge
compression for speeding up queries to propositional Horn expert systems is
equivalent to HM.
HM is a computationally difficult problem: it is known to be NP-hard even if the
input is restricted to cubic Horn CNFs. On the other hand, there are two subclasses of
Horn CNFs for which HM is known to be solvable in polynomial time: acyclic and
quasi-acyclic Horn CNFs. In this talk we introduce a new class of Horn CNFs which
properly contains both of the known classes and describe a polynomial time HM
algorithm for this new class.
3. Alexander Zadorojniy, Guy Even and Adam Shwartz, "A strongly polynomial
algorithm for controlled queues"
We consider the problem of computing optimal policies of finite-state, finite-action
Markov Decision Processes (MDPs). A reduction to a continuum of constrained
MDPs (CMDPs) is presented such that the optimal policies for these CMDPs
constitute a path in a graph defined over the deterministic policies. This path
contains, in particular, an optimal policy of the original MDP. We present an
algorithm based on this new approach that finds this path and thus an optimal policy.
In the general case this path might be exponentially long in number of states and
actions. We prove that the length of this path is polynomial if the MDP satisfies a
coupling property. Thus we obtain a strongly polynomial algorithm for MDPs that
satisfy the coupling property. We prove that discrete time versions of controlled
M/M/1 queues induce MDPs that satisfy the coupling property. The only previously
known polynomial algorithm for controlled M/M/1 queues in the expected average
cost model is based on linear programming (and is not known to be strongly
polynomial). Our algorithm works both for the discounted and expected average cost
models, and the running time does not depend on the discount factor.
Track 2: Scheduling 1:
1.
Bezalel Gavish, "Batch sizing and scheduling in serial multistage production
system"
The authors propose a procedure to model and determine the optimal batch sizes and
their schedule in a serial multistage production process with constant demand rate for
the final product. The model takes into consideration the batch setup time in each
stage, the processing times of a batch as a function of batch size and production stage.
It considers the holding and shortage costs of the final product, and the setup and
processing costs at each stage of the production process. The objective is to maximize
the net profit per unit of time derived from selling the final products. We report on
extensive computational experiments that tested the impact of different factors on the
objective function and the optimal batch size. A feature unique to this paper is the
concentration on multistage serial systems consisting of hundreds of stages in the
production process, in such systems a batch can spend from one to eight months in the
production process. Batch sizing and scheduling has a great impact on the net
revenues derived from the production facility. Due to the tight tolerances of the
production process which are computer controlled, setup and processing times are
deterministic.
2. Yaron Leyvand, Dvir Shabtay, George Steiner and Liron Yedidsion, "Just-intime scheduling with controllable processing times on single and parallel
machines"
We study bicriteria scheduling problems with controllable processing times on single
and parallel machines. Our objectives are to maximize the weighted number of jobs
that are completed exactly at their due date and to minimize the total resource
allocation cost. We consider four different models for treating the two criteria. We
prove that three of these problems are NP-hard even on a single machine, but
somewhat surprisingly, the problem of maximizing an integrated objective function
can be solved in polynomial time even for the general case of a fixed number of
unrelated parallel machines. For the three NP-hard versions of the problem, with a
fixed number of machines and a discrete resource type, we provide a pseudopolynomial time optimization algorithm, which is converted to a fully polynomial
time approximation scheme to find a Pareto optimal solution.
3. Gur Mosheiov and Assaf Sarig, "Scheduling a maintenance activity to minimize
total weighted completion-time"
We study a single machine scheduling problem. The processor needs to go through a
maintenance activity, which has to be completed prior to a given deadline. The
objective function is minimum total weighted completion time. The problem is proved
to be NP-hard, and an introduction of a pseudo-polynomial dynamic programming
algorithm indicates that it is NP-hard in the ordinary sense. We also present an
efficient heuristic which is shown numerically to perform well.
15:25-16:15 Parallel sessions:
Track 1: Supply chain:
1. Tadeusz Sawik, "A two-objective mixed integer programming approach for
supply chain scheduling"
The integrated and hierarchical approaches based on mixed integer programming are
proposed for a bi-objective coordinated scheduling in a customer driven supply chain.
The supply chain consists of multiple suppliers (manufacturers) of parts and a single
producer of finished products. Given a set of customer orders, the problem objective
is to determine a coordinated schedule for the manufacture of parts by each supplier,
for the delivery of parts from each supplier to the producer, and for the assignment of
orders to planning periods at the producer, such that a high revenue or a high
customer service level is achieved and the total cost of holding supply chain inventory
is minimized.
The proposed two approaches are described below.
1. Integrated (simultaneous) approach.
The coordinated schedules for
customer orders, for manufacturing of parts and for supply of parts are
determined simultaneously to achieve a minimum number of tardy orders or a
minimum lost revenue due to tardiness or full rejection of orders, at a
minimum cost of holding total supply chain inventory of parts and finished
products.
2. Hierachical (sequential) approach.
First, the order acceptance/due date
setting decisions are made to select maximal subset of orders that can be
completed by requested due dates, and for the remaining orders delayed due
dates are determined to minimize the lost revenue owing to tardiness or
rejection of the orders. The due dates must satisfy capacity constraints and are
considered to be deadlines at the order scheduling level. Next, order deadline
scheduling is performed to meet all committed due dates and to minimize cost
of holding finished product inventory of orders completed before the
deadlines. Finally, scheduling of manufacturing and supply of parts,
coordinated with the schedule for orders is accomplished to meet demand for
parts at a minimum cost of part supply and inventory holding.
Numerical examples modeled after a real-world scheduling in a customer driven
supply chain of high-tech products are presented and some computational results are
reported to compare the two approaches. To determine subsets of non-dominated
solutions for the integrated approach, in the computational experiments the weightedsum program is compared with the Tchebycheff program.
2. Grzegorz Pawlak, "Multi-layer agent scheduling and control model in the car
factory"
The complex planning and production tasks in the car production factory will generate
the necessity to apply complicated scheduling and control systems. The goal of the
work is to have the production in balance, without breaks and without having too
many parts in a stock. The production process is controlled and measured by taking
into account many coefficients. The inventory level, the buffers occupation,
production flow, deadlines, sequence quality, failures etc. Building the global
optimization system is difficult and usually such a system is not considering all
constraints and dependencies. The alternative is to build the model with the
distributed local agents – schedulers and control units fast adapting to the changing
circumstances. The disturbance of the process caused by the demand changes or break
downs in the lines should lead to the appropriate reaction of the scheduling and
control system. In the fast changing environment the intelligent and self adopted
scheduling and control system may be the advantage in the production optimization
process.
In the situations where is a need to set the production speed or the synchronization of
production line in multi-line production system the agent driven control could be the
reasonable alternative. Additionally, the usage and intelligent control of buffers and
switches can improve the production efficiency. In every car there are thousands of
parts to be linked and that is why such production system, can be regarded as one of
most complicated and other segment companies can follow that application example.
So the model is not restricted only to the car factories.
The presented model is taking into account the multi – layer control agent system.
There are agents to play specific roles. There is the control, monitoring and
visualization, scheduling agents. Each of them has defined the execution domain and
the activity area. The configuration and the management of such agent systems have
been taken into account. The classical scheduling models and algorithms have been
incorporated into the bigger view of the production
Summarizing, there is a need to have a self adopting production control system, which
will be able to keep production with appropriate speed available for the demand
driven production plan and according to the current state of production facilities and
resources. In response to this problem, we would like to create the multi- line, self
adopting production control system, which will consists of high-level control
software, communication protocols and production line controllers behavior
scenarios, prepared for different modes and situations.
Track 2: Networks:
1. Oren Ben-Zwi, Danny Hermelin, Daniel Lokshtanov and Ilan Newman, "An
exact almost optimal algorithm for target set selection in social networks"
The Target Set Selection problem proposed by Kempe, Kleinberg, and Tardos, gives a
nice clean combinatorial formulation for many problems arising in economy,
sociology, and medicine. Its input is a graph with vertex thresholds, the social
network, and the goal is to find a subset of vertices, the target set, that ``activates" a
prespecified number of vertices in the graph. Activation of a vertex is defined via a
so-called activation process as follows: Initially, all vertices in the target set become
active. Then at each step i of the process, each vertex gets activated if the number of
its active neighbors at iteration i-1 exceeds its threshold. The activation process is
``monotone" in the sense that once a vertex is activated, it remains active for the
entire process.
Unsurprisingly perhaps, Target Set Selection is NPC. More surprising is the fact that
both of its maximization and minimization variants turn out to be extremely
hard to approximate, even for very restrictive special cases. The only known case for
which the problem is known to have some sort of acceptable worst-case solution is the
case where the given social network is a tree and the problem becomes polynomialtime solvable. In this paper, we attempt at extending this sparse landscape of tractable
instances by considering the treewidth parameter of graphs. This parameter roughly
measures the degree of tree-likeness of a given graph, e.g. the treewidth of a tree is 1,
and has previously been used to tackle many classical NPhard problems in the
literature.
Our contribution is twofold: First, we present an algorithm for Target Set Selection
running in nO(w) time, for graphs with n vertices and treewidth bounded by w. The
algorithm utilizes various combinational properties of the problem; drifting somewhat
from standard dynamic-programming algorithms for small treewidth graphs.
Also, it can be adopted to much more general settings, including the case of directed
graphs, weighted edges, and weighted vertices. On the other hand, we also show that
it is highly unlikely to find an no(\sqrt{w}) time algorithm for Target Set Selection, as
this would imply a sub-exponential algorithm for all problems in SNPclass. Together
with our upper bound result, this shows that the treewidth parameter determines the
complexity of Target Set Selection to a large extent, and should be taken into
consideration when tackling this problem in any scenario.
2. André Amaral, Alberto Caprara, Juan José Salazar González and Adam
Lechford, "Lower bounds for the minimum linear arrangement of a graph"
Minimum Linear Arrangement is a classical basic combinatorial optimization
problem from the 1960s that turns out to be extremely challenging in practice. In
particular, for most of its benchmark instances, even the order of magnitude of the
optimal solution value is unknown, as testified by the surveys on the problem that
contain tables in which the best known solution value has many more digits than the
best known lower bound value. Since, for most of the benchmark instances, finding a
provably optimal solution appears to be completely out of reach at the moment, in this
paper we propose a linear-programming based approach to compute lower bounds on
the optimum. This allows us, for the first time, to show that the best known solutions
are indeed not far from optimal.
16:45-17:45 Plenary talk – Maurice Queyranne "Structural and algorithmic
properties for parametric minimum cuts"
We consider the minimum s-t-cut problem in a network with parametrized arc
capacities. Classes of this parametric problem have been shown to enjoy the nice
structural property that minimum cuts are nested and, following the seminal work of
Gallo, Grigoriadis and Tarjan (1989), the nice algorithmic property that all minimum
cuts can be computed in the same asymptotic time as a single minimum cut.
We present a general framework for parametric minimum cuts that extends and
unifies such results.
We define two conditions on parametrized arc capacities that are necessary and
sufficient for (strictly) decreasing differences of the parametric cut function.
Known results in parametric submodular optimization then imply the structural
property.
We show how to construct appropriate flow updates in linear time under the above
conditions, implying that the algorithmic property also holds under these conditions.
We then consider other classes of parametric minimum cut problems, without
decreasing differences, for which we establish the structural and/or the algorithmic
property, as well as other cases where nested minimum cuts arise.
This is joint work with Frieda Granot and S. Thomas McCormick (Sauder School of
Business at UBC) and Fabio Tardella (Universitá La Sapienza, Rome)
17:55-19:10 Parallel sessions:
Track 1: Scheduling 2:
1. Jan Pelikán, " Hybrid flow shop with adjustment"
The subject of this paper is a flow-shop based on a case study aimed at the
optimisation of ordering production jobs in mechanical engineering, in order to
minimise the overall processing time, the makespan.
There is a given set of production jobs to be processed by the machines installed on
the shop floor. A job is a product batch for a number of units of given product
assigned to processing on the machine. Each job is assigned to a certain machine,
which has to be adjusted by an adjuster. This worker adjusts all of the machines
installed on the floor but at a given time, he can adjust only one machine, i.e. he
cannot adjust more machines simultanously. Each adjusted machine is supposed to be
capable of immediate starting to process the job for which it has been adjusted. No job
processing is allowed to be broken (i.e., intermittent processing is not admissible), and
each machine is able to process only one job at a time. The processing time
minimisation problem is aimed at the reduction of the waiting times of both the
machines for the adjuster and of the adjuster for a machine to be adjusted. After
completing a job on the machine, the machine has to wait if the adjuster is finishes
adjusting another machine. If all machines are processing jobs, the adjuster has to
wait. A solution is represented by the order of the jobs in which the adjuster adjusts
the respective machines. This order also determines the order of the machines to be
adjusted by the adjuster. At the same time, the job ordering generates the order of the
jobs on the machines when several jobs are assigned to the same machine. In the
literature, hybrid flow-shop is defined as a problem of processing jobs which consists
of two or more stages, with one or more processors at each stage. Each of the jobs to
be processed consists of two or more tasks and each task is processed within its own
stage. The jobs are non-preemptable and each subsequent stage is only started after
the processing of the previous stage is completed. A hybrid flow-shop as described
here further assumes that the following job at the first stage can be processed
immediately upon completion of the preceding job at the first stage. Therefore, the
problem of this case study can be formulated as a two-stage hybrid flow-shop, in
which the first stage is represented by the work of an adjuster, who is viewed as the
only one processor at the first stage. The second stage is represented by the machines
on the shop floor, viewed as parallel processors; however, jobs are uniquely assigned
to these processors. The objective function is defined as the makespan value, i.e., the
overall processing time spent on all jobs at both stages. Contrary of the hybrid flowshop defined in literature, there are two differences for two consecutive jobs here:
a) If both jobs are assigned to the same machine, i.e., the same second-stage
processor, the second job's first stage can only be started upon completion of the first
job's second stage, when the machine is available;
b) If each of the two consecutive jobs is assigned to a different machine as a secondstage processor, the second job's first stage can be started immediately upon
completion of the first job's first stage.A mathematical model is proposed, a heuristic
method is formulated, and the NP hardness of the problem, called a "hybrid flow-shop
with adjustment," is proved.
2. Alexander Lazarev, "An approximation method for estimating optimal value of
minimizing scheduling problems"
3. Alexander Kvaratskhelia and Alexander Lazarev, "An algorithm for total
weighted completion time minimization in preemptive equal job length with release
dates on a single machine"
We consider the minimizing total weighted completion time in preemptive equal job
length scheduling problem on a single machine with release dates. We propose a
polynomial time algorithm that solves the problem. Before this paper, the problem is
known to be open
(http://www.lix.polytechnique.fr/~durr/OpenProblems/1_rj_pmtn_pjp_sumWjCj/).
We refer to the total weighted completion time problem (without release dates,
preemption and arbitrary job length) that can be solved in polynomial time using
Smith’s rule.
Our algorithm is also based on the Smith’s rule, which is applied to solve a subproblem with purpose to find an optimal processing order within a time interval
between two adjacent release dates. An optimal schedule is being found starting from
the least release date. On each step we construct a sub-problem of jobs, which is
already available wince corresponding release date. We find an optimal processing
order of these jobs using Smith’s rule, and then we switch to the next release date.
We propose a number of properties of the optimal schedule, which prove the
optimality of the schedule constructed by Algorithm in polynomial time.
Track 2: Optimization 1:
1. Michael Katz and Carmel Domshlak, "Pushing the envelope of abstraction-based
admissible heuristics"
The field of automated, domain-independent planning seeks to build general-purpose
algorithms enabling a system to synthesize a course of action that will achieve certain
goals. Such algorithms perform reachability analysis in large-scale state models that
are implicitly described in a concise manner via some intuitive declarative language.
And though planning problems have been studied since the early days of Artificial
Intelligence research, recent developments (and, in particular, recent developments in
planning as heuristic search) have dramatically advanced the field, and also
substantially contributed to some related fields such as software/hardware
verification, control, information integration, etc.
The difference between various algorithms for planning as heuristic search is mainly
in the heuristic functions they define and use. Most typically, an (admissible) heuristic
function for domain-independent planning is defined as the (optimal) cost of
achieving the goals in an over-approximating abstraction of the planning problem in
hand. Such an abstraction is obtained by relaxing certain constraints that are present in
the specification of the real problem, and the desire is to obtain a provably poly-time
solvable, yet informative abstract problem. The main questions are thus:
1.What constraints should we relax to obtain such an effective over-approximating
abstraction?
2. How should we combine information provided by multiple such abstractions?
In this work we consider both these questions, and present some recent formal results
that help answering these questions (sometimes even to optimality). First, we consider
a generalization of the popular ``pattern database'' (PDB) homomorphism abstractions
to what is called ``structural patterns''. The basic idea is in abstracting the problem in
hand into provably tractable fragments of optimal planning, alleviating by that the
constraint of PDBs to use projections of only low dimensionality. We introduce a
general framework for additive structural patterns based on decomposing the problem
along certain graphical structure induced by it, suggest a concrete non-parametric
instance of this framework called fork-decomposition, and formally show that the
admissible heuristics induced by the latter abstractions provide state-of-the-art worstcase informativeness guarantees on several standard domains. Specifically, we
describe a procedure that takes a classical planning task, a forward-search state, and a
set of abstraction-based admissible heuristics, and derives an optimal additive
composition of these heuristics with respect to the given state. Most importantly, we
show that this procedure is polynomial-time for arbitrary sets of all known to us
abstraction-based heuristics.
2. Roberto Battiti and Paolo Campigotto, "Brain-machine optimization: learning
objective functions by interacting with the final user"
As a rule of thumb, ninety percent of the problem-solving effort in the real world is
spent on defining the problem, on specifying in a computable manner the function to
be optimized. After this modeling work is completed, optimization becomes in certain
cases a commodity. The implication is that much more research effort should be
devoted to design supporting techniques and tools to help the final user, often without
expertise in mathematics and in optimization, to define the function corresponding to
his real objectives. Machine learning can help, after the important context of
interfacing with a human user to provide feedback signals is taken into account, and
after the two tasks of modeling and solving are integrated into a loop so that an initial
rough definition can be progressively refined by bridging the representation gap
between the user world and the mathematical programming world.
To consider a more specific case, many real world optimization problems are
typically multi-objective optimization (MOO) problems. Efficient optimization
techniques have been developed for a wide range of MOO problems but, as
mentioned, asking a user to quantify the weights of the different objectives before
seeing the actual optimization results is in some cases extremely difficult. If the final
user is cooperating with an optimization expert, misunderstanding between the
persons may arise. This problem can become dramatic when the number of the
conflicting objectives in MOO increases. As a consequence, the final user will often
be dissatisfied with the solutions presented by the optimization expert. The
formalization of the optimization problem may be incomplete because some of the
objectives remain hidden in the mind of the final user.
Although providing explicit weights and mathematical formulas can be difficult for
the user, for sure he can evaluate the returned solutions. In most cases, the strategy to
overcome this situation consists of a joint iterative work between the final user and
the optimization expert to change the definition of the problem itself. The
optimization tool will then be re-executed over the new version of the problem.
We design a framework to automatically learn the knowledge that cannot be clearly
stated by the final user, i.e., the weights of the ``hidden" objectives. Each run of an
optimization tool provides a set of non-dominated solutions. The learning process we
investigate is based on the final user evaluation of the solution presented. Even if the
final user cannot state in a clear and aware manner the unsatisfied objectives, he can
select his favorite solutions among the ones provided by the optimization expert, after
the results are appropriately clustered and a suitably small number of questions is
answered.
In particular, our framework is based on the Reactive Search Optimization (RSO)
technique, which advocates the adoption of learning mechanisms as an integral part of
heuristic optimization schemes for solving complex optimization problems. The
ultimate goal of RSO is the complete elimination of the human intervention in the
parameters tuning process of heuristic algorithms. Our framework replaces the
optimization expert in the definition of the optimization task, by directly learning the
``hidden" objective functions from the final user evaluation of the solutions provided
by the optimization tools. The theoretical framework and some experimental results
will be presented in the extended version of this work.
3.
Lukasiak Piotr, Blazewicz Jacek, David Klatzmann, " CompuVac* development and standardized evaluation of novel genetic vaccines".
Recombinant viral vectors and virus-like particles are considered the most promising
vehicles to deliver antigens in prophylactic and therapeutic vaccines against infectious
diseases and cancer. Several potential vaccine designs exist but their cost-effective
development cruelly lacks a standardized evaluation system. On these grounds,
CompuVac is devoted to (i) rational development of a novel platform of genetic
vaccines and (ii) standardization of vaccine evaluation.
CompuVac assembles a platform of viral vectors and virus-like particles that are
among today’s most promising vaccine candidates and that are backed up by the
consortium’s complementary expertise and intellectual property, including SMEs
focusing on vaccine development.
CompuVac recognizes the lack of uniform means for side-by-side qualitative and
quantitative vaccine evaluation and will thus standardize the evaluation of vaccine
efficacy and safety by using “gold standard” tools, molecular and cellular methods in
virology and immunology, and algorithms based on genomic and proteomic
information.
“Gold standard” algorithms for intelligent interpretation of vaccine efficacy and safety
will be built into Compuvac’s interactive “Genetic Vaccine Decision Support
System”, which should generate (i) vector classification according to induced immune
response quality, accounting for gender and age, (ii) vector combination counsel for
prime-boost immunizations, and (iii) vector safety profile according to genomic
analysis.
Main objectives of CompuVac are:
•
to standardize the qualitative and quantitative evaluation of genetic vaccines
using defined “gold standard” antigens and methods
•
to rationally develop a platform of novel genetic vaccines using genomic and
proteomic information, together with our gold standards
•
to generate and make available to the scientific community a “tool box” and
an “interactive database” allowing to comparatively assess future vaccines to be
developed with our gold standards
Vector platform used in CompuVac is made of both viral vectors and VLPs that are
representative of and considered among the best in their class. Some, like the
adenoviruses, are already in clinical development. Within this consortium, we will
perform rational vector improvements and test rational prime boost combinations, all
guided by our comparative evaluation system. This should generate safer and more
efficient vectors, or vector combinations, for clinical development.
The consortium will generate an interactive platform called GeVaDS (Genetic
Vaccine Decision Support system). The platform will contain formatted data related
to defined gold standard antigens and methods used to assess immune responses i.e.
newly acquired results as well as algorithms allowing the intelligent comparison of
new vectors to previously analyzed ones.
CompuVac aims at making GeVaDS accessible to any researcher developing genetic
vaccines. Retrieving previously generated or introducing newly acquired results
obtained with validated approved methods, should allow any researcher to rationally
design and improve his or her vaccine vector as well as to comparatively assess its
efficacy and potential. Internet access:
http://www.compuvac.org (CompuVac),
http://gevads.cs.put.poznan.pl (GeVaDSs).
Monday
9:00-10:00 Plenary talk – Yefim Dinitz "Efficient algorithms for AND/OR (maxmin) scheduling problems on graphs"
The talk considers scheduling with AND/OR precedence constraints. The events are
represented as the vertices of a non-negatively weighted graph G=(V,E,d). The
precedence relation between them is given by its edges, and the delays by edge
weights. An event of type AND (max) should be scheduled anywhen after all its
preceding events, with the corresponding delays after them. An event of type OR
(min) should be scheduled anywhen after at least one of its preceding events, with the
corresponding delay after it. The early schedule is in question. This problem is a maxmin generalization of PERT. Besides, it may be considered as a mixture of the
Shortest Path and PERT problems.
We first consider the case when zero weight cycles are absent in G. Surprisingly,
some natural mixture of the classic Dijkstra and PERT algorithm works for it.
It solves the problem in time of two those algorithms, which is almost linear in the
graph size. After that, we present an O(|V||E|) algorithm solving the problem in the
general case, with zero weight cycles allowed.
We describe also a quite special history of the research on the AND/OR scheduling
problem. It began from a paper of D. Knuth in late 70th (presenting the algorithm, but
not the problem), went over particular cases and proofless publications, and involved
also names of Moehring, Adelson-Velsky, Levner, the lecturer, and others. Currently,
the first full paper on its general case is in preparation.
10:30- 11:45 Parallel sessions:
Track 1: Metaheuristics:
1. Johan Oppen, "Parametric models of local search progression"
Algorithms that search for good solutions to NP-hard combinatorial optimization
problems present a trace of current best objective values over time. The progression of
objective function values for best solutions found is reasonably modeled as stochastic
because the algorithms are often stochastic, and even when they are not, the
progression of the search varies with each instance in unpredictable ways. Some
characteristics are common to most searches: new bests are found quickly early in the
search and not as quickly later on.
We describe parametric models of this progression that are both interesting as ways to
characterize the search progression in a compact way and useful as means of
predicting search behavior. This is in contrast to non-parametric models that estimate
the probability that the search will achieve a value better than some given threshold in
a given amount of time. Both types of models have value, but parametric models offer
the promise of greater predictive power. For practical purposes, it is useful to be able
to use small instances to estimate parameters for models that can then be used to
predict the time trace performance on large instances.
2. Gregory Gutin and Daniel Karapetyan, "Local search heuristics for the
multidimensional assignment problem"
The Multidimensional Assignment Problem (MAP) (abbreviated s-AP in the case of s
dimensions) is an extension of the well-known assignment problem. The most
studied case of MAP is 3-AP, though the problems with larger values of s have also a
number of applications. We consider several known and new MAP local search
heuristics for MAP as well as their combinations and simple metaheuristics.
Computational experiments with three instance families are provided and discussed.
As a result, we select dominating local search heuristics. One of the most interesting
conclusions is that combination of two heuristics may yield a superior heuristic with
respect to both solution quality and the running time.
3. José Brandão, "A tabu search algorithm for the open vehicle routing problem
with time windows"
The problem studied here, the open vehicle routing problem with time windows
(OVRPTW), is different from the vehicle routing problem with time windows in that
the vehicles do not return to the distribution depot after delivering the goods to the
customers. We have solved the OVRPTW using a tabu search algorithm with
embedded local search heuristics, which take advantage of the specific properties of
the OVRP. The performance of the algorithm is tested using a large set of benchmark
problems.
Track 2: Networks:
1. Silvano Martello, "Jacobi, Koenig, Egerváry and the roots of combinatorial
optimization"
It is quite well-known that the first modern polynomial-time algorithm for the
assignment problem, invented by Harold W. Kuhn half a century ago, was christened
the "Hungarian method" to highlight that it derives from two older results, by Koenig
(1916) and Egerváry (1931). A recently discovered posthumous paper by Jacobi
(1804-1851) contains however a solution method that appears to be equivalent to the
Hungarian algorithm. A second historical result concerns a combinatorial
optimization problem, independently defined in satellite communication and in
scheduling theory, for which the same polynomial-time algorithm was independently
published thirty years ago by various authors. It can be shown that such algorithm
directly implements another result by Egerváry, which also implies the famous
Birkhoff-von Neumann theorem on doubly stochastic matrices.
2. Shoshana Anily and Aharona Pfeffer, " The uncapacitated swapping problem on
a line and on a circle"
We consider the uncapacitated swapping problem on a line and on a circle. Objects of
m types, which are initially positioned at n workstations on the graph, need to be
rearranged in order to satisfy the workstations’ requirements. Each workstation
initially contains one unit of a certain object type and requires one unit of possibly
another object type. We assume that the problem is balanced, i.e., the total supply
equals the total demand for each of the object types separately. A vehicle of unlimited
capacity is assumed to ship the objects in order to fulfill the requirements of all
workstations. The objective is to compute efficiently a shortest route such that the
vehicle can accomplish the rearrangement of the objects while following this route,
given designated starting and ending workstations on a line, or the location of a depot
on a circle.
We propose polynomial-time exact algorithms for solving the problems: an O(n)-time
algorithm for the linear track case, and an O(n2)-time algorithm for the circular track
case.
3. Susann Schrenk, Van-Dat Cung and Gerd Finke, "Revisiting the fixed charge
transportation problem"
The classical transportation problem is a special case of a minimum cost flow
problem with the property that the nodes of the network are either supply nodes or
demand nodes. The sum of all supplies and the sum of all demands are equal, and a
shipment between a supply node and a demand node induces a linear cost. The aim is
to determine a least cost shipment of a commodity through the network in order to
satisfy demands from available supplies. This classical transportation problem was
first presented by Hitchkock in 1941. In 1951 Dantzig proposed a standard
formulation as a linear program along with a first implementation of the simplex
method to solve it. This problem is polynomial.
The transportation problem including a fixed cost associated to each transportation
arc, named fixed charge transportation problem, is NP-Hard. This problem was first
introduced by Hirsch and Dantzig in 1954 and has largely been investigated in the
literature, Dantzig and Hirsch 1968, Sharma 1977, Balinski 1961. Most of papers until
then focused on approximation schemes to solve the fixed charge transportation
problem.
We study the special case, with no variable cost and the same fixed cost on all arcs.
This is in a way the simplest form of this problem, but we can show that this problem
remains NP-Hard. Also, minimizing the transportation costs is in this case equivalent
to having the maximum number of arcs on which there is no transport. That is
equivalent to finding a solution with a maximum degree of degeneracy. Since the
basic solutions are trees, solving the problem is equivalent to finding the tree with
maximum degree of degeneracy. Degeneracy occurs if one has a subset of the supply
nodes and a subset of the demand nodes whose total demand and total supply are
equal. We refer to this property as equal-subsum-problem. The existence of
degeneracy is an NP-Complete problem. Finding the maximum degree of degeneracy
is therefore also NP-Complete.
A new result in this context is the following: Finding the maximum degree of
degeneracy remains NP-Complete even if all equal subsums are given. The proof of
this theorem involves the maximum stable problem and the set packing problem.
12:00-12:50 Parallel sessions:
Track 1: Optimization 2:
1. Gur Mosheiov and Assaf Sarig, "Scheduling and due-date assignment problems
with job rejection"
Scheduling with rejection reflects a very common scenario, where the scheduler may
decide not to process a job if it is not profitable. We study the option of rejection in
several popular and well-known scheduling and due-date assignment problems. A
number of settings are considered: due-date and due-window assignment problems
with job-independent costs, a due-date assignment problem with job-dependent
weights and unit jobs, minimum total weighted earliness and tardiness cost with jobdependent and symmetric weights (known as TWET), and several classical
scheduling problems (minimum makespan, flow-time, earliness-tardiness) with
position-dependent processing times. All problems (excluding TWET) are shown to
have a polynomial time solution. For the (NP-hard) TWET, a pseudo-polynomial time
dynamic programming algorithm is introduced and tested numerically.
2. Ephraim Korach and Michal Stern, "TDIness in the clustering tree problem"
We consider the following problem: Given a complete graph G=(V,E) with a weight
on every edge and a given collection of subsets of V, we have to find a minimum
weight spanning tree T such that each subset of the vertices in the collection induces a
subtree in T. This problem arises and has motivation in the field of communication
networks. Consider the case where every subset is of size at most three. We present a
linear system that defines the polyhedron of all the feasible solutions. We prove that
this system is Total Dual Integrality (TDI). We present a conjecture for the general
case, where the size of each subset is not restricted to be less or equal to three. This
conjecture generalizes the above results.
Track 2: Convex combinatorial optimization problems:
1. Tiit Riismaa, "Application of convex extension of discrete-convex functions in
combinatorial optimization"
Convex extension of discrete-convex functions enables to adapt effective methods of
the convex programming for solving some discrete-convex programming problems of
hierarchy optimization. The n – dimensional real-valued function is called discreteconvex if the inequality of Jensen is valid for all convex combinations of (n+1) –
elements from the domain of definition. The use of all n+1 elements convex
combinations follows from the well-known theorem of Caratheodory .
The convex extension of given function is the majorant convex function not
exceeding the given function. Theorem. The function f can be extended to convex
function on conv X if f is discrete-convex on X.
The graph of a discrete-convex function is a part of the graph of a convex function.
The convex extension is so called point-wise maximum over all linear functions not
exceeding the given function. The convex function of a discrete-convex function is a
piecewise linear function. Each discrete-convex function has a unique convex
extension. The class of discrete-convex functions is the largest one to be extended to
the convex functions. A class of iteration methods of local searching is developed. On
each step of the iteration the calculation of the value of objective function is required
only on some vertices of some kind of unit cube. Many discrete or finite hierarchical
structuring problems can be formulated mathematically as a multi-level partitioning
procedure of a finite set of nonempty subsets. This partitioning procedure is
considered as a hierarchy where to the subsets of partitioning correspond nodes of
hierarchy and the relation of containing of subsets define the arcs of the hierarchy.
The feasible set of structures is a set of hierarchies corresponding to the full set of
multi-level partitioning of given finite set. Each tree from this set is represented by a
sequence of Boolean matrices, where each of these matrices is an adjacency matrix of
neighboring levels. The described formalism enables to state the reduced problem as a
two-phase mutually dependent optimization problem. Variable parameters of the inner
minimization problem are used for the description of connections between adjacent
levels. Variable parameters of the outer minimization problem are used for the
presentation of the number of elements on each level. The two-phase statement of
hierarchy optimization problem guarantees the possibility to extend the discreteconvex objective function to the convex function and enables to apply algorithm of
local searching for finding the global optimum. The approach is illustrated by a multilevel production system example.
2. Boaz Golany, Moshe Kress, Michal Penn and Uriel Rothblum, "Resource
allocation in temporary advantage competitions"
We consider a race between two competitors that develop competing products where
the advantage that is achieved by one competitor is limited in time until the opponent
is developing a new (better) product. We model the problem as network optimization
with side constraints and obtain some insights. We further briefly discuss the
stochastic case where the problems are modeled as (convex) non-linear optimization
problems.
Tuesday
9:00-10:15 Parallel sessions:
Track 1: Scheduling 3:
1. Liron Yedidsion, "Computational analysis of two fundamental problems in
MRP"
In this research we establish the computational complexity for two of the most basic
problem in the field of Material Requirement Planning (MRP). In particular, the
minimization of the total cost in the multi-echelon Bill of Material (BOM) system and
in the assembly system. Although both problems have been vastly studied since the
early 50s of the 20th century, their computational complexity has remained an open
question heretofore. We prove that both problems are strongly NP-hard, where the
minimization of the total cost in the multi-echelon BOM system is strongly NP-hard
even for the case where the Lead-Time of all the components is zero.
2. Sergey Sevastyanov and Bertrand M.T. Lin, "Efficient algorithm of generating all
optimal sequences for the two-machine Johnson problem"
For the classical two-machine Johnson problem we make it our aim to generate the set
of all optimal sequences of jobs (with respect to the minimum makespan objective),
assuming that on the set of those optimal sequences we could wish to optimize some
other (secondary) criterion.
Clearly, this purpose could be attained by enumerating all n! sequences of n jobs, and
this is the best possible bound for some instances (for example, for those consisting of
identical jobs). Yet for the majority of problem instances (having only a few optimal
solutions) this way cannot be recognized as acceptable, in view of its excessive timeconsumption.
We say that an algorithm pursuing the above mentioned purpose is efficient if its
running time can be estimated as O(P(|I|) |\Pi_{OPT}|), where P(|I|) is a polynomial of
the input size, while \Pi_{OPT} is the set of all optimal sequences of jobs. In other
words, if obtaining every optimal sequence requires (in average) at most a polynomial
time.
Such an efficient algorithm was designed in our paper. Its running time can be
estimated as O(n\log n|\Pi_{OPT}|). Therefore, in average we spend for generating
each new optimal sequence the same time as Johnson spent for his unique optimal
sequence. The algorithm is based on several lemmas containing basic properties of
optimal sequences.
3. Alexander Lazarev, "An approximate method for solving scheduling problems"
We consider the scheduling problems α|β|Fmax: A set of n jobs J1,…,Jn with release
dates r1,…,rn, processing times p1,…,pn and due dates d1,…,dn has to be scheduled on
a single or many machines. The job preemption is not allowed. The goal is to find a
schedule that minimizes the regular function F(C1,…,Cn), that Cj is the job j
completion time. We have suggest an approximation scheme to find approximate
optimal value of the objective function.
The approximation scheme: Let we have a instance A={rj,pj,dj} of our scheduling
problem. And for solving the instance we haven't sufficient computer resource. The
idea of the approach consists: we draw a line in 3n-dimensional space through a
point (instance) A. For example, {(rj,γpj,dj)|j\in N}, where γ \in [0,+ ∞). At γ =1 we
have initial instance A. Then on the given line "the reasonable" interval for Γ 1 and Γ2,
Γ1 < γ < Γ2, Γ1 < 1 < Γ2, is chosen. Usually at boundary points (instances) of an
interval the solution of scheduling problem is found for comprehensible time. Let H
be "limiting top" size of computer possibilities. For example, we choose values γ 1,…,
γk as roots of Chebyshev's polynomial on the interval $[ Γ1, Γ2], as they "nestle" on
the interval edges where we can find the solutions of the problem. For the instances
corresponding γ1,…, γk for which enough computing possibilities we find value of the
objective function F1,…,Fk. Then through points (γ1,F1),…, (γk,Fk) we construct
Lagrange's polynomial. Then the criterion function error Δ in our initial instance (with
γ =1) will not exceed size Δ ≤ C |(1- γ1)…(1- γk)|/(k+1)!, where C is constant. As a
result we do not construct the schedule of the solution of the problem, we only
estimate value of objective function. Than it is more points k, and the it is less
distance between Γ2 and Γ1 the less size Δ.
Track 2: Heuristics 1:
1. Mark Sh. Levin, "Combinatorial optimization problems in system configuration
and reconfiguration"
In recent years, systems configuration and reconfiguration problems are used as a
special significant part of design and management for many applications (e.g.,
software, hardware, manufacturing, communications, supply chain systems, solving
strategies, modular planning, combinatorial chemistry).
In the paper several basic system configuration/reconfiguration problems are
examined: (i) searching for (selection of) a set (structure) of system components, (ii)
searching for a set of compatible system components, (iii) allocation of system
components, (iv) design of system hierarchy, and (v) reconfiguration of a system
(e.g., change of system structure or system hierarchy).
The following combinatorial optimization problems are considered as underlying
models (including multicriteria formulations): problem of representatives,
multicriteria multiple choice problem, multicriteria allocation problem, graph coloring
and recoloring problems, morphological clique problem (with compatibility of system
components), multipartite clique or clustering of multipartite graph, hierarchical
clustering, and minimal spanning trees (including Steiner tree) problems. Solving
approaches and applications are discussed.
2. Aleksandra Swiercz, Jacek Blazewicz, Marek Figlerowicz, Piotr Gawron, Marta
Kasprzak, Darren Platt and Łukasz Szajkowski, "A new method for assembling of
DNA sequences"
Genome sequencing is the process of recovering a DNA sequence of a genome. It is
well known from its high complexity both on biological and computational levels.
The process is composed of a few phases: sequencing, assembling and finishing.
In the first phase one can obtain short DNA sequences of length up to a few hundreds
of nucleotides (in case of Sanger method). In case of new approaches to DNA
sequencing (e.g. 454 sequencing) shorter DNA sequences are obtained with relatively
good qualities and in short time. As the output of the 454 sequencing one gets
together with sequences (of length around 100-200 nucleotides), qualities, i.e. the
confidence rates of each nucleotide.
In the next phase, the assembling, short DNA sequences coming from the sequencing
phase are combined into a longer sequence or contiguous sequences, called contigs. In
the last phase, the finishing, which can be omitted in case of small genomes, the
contigs are put into the proper place of the chromosome.
The novel heuristic algorithm, SR-ASM, has been proposed for the assembly
problem, which deals well with data coming from the 454 sequencing. The algorithm
is based on a graph model. In the graph the input DNA sequences are on vertices, and
arcs connect two vertices, if two respective sequences overlap. The algorithm is
composed of three parts. In the first part the graph is constructed. At the beginning, a
fast heuristics looks for the pairs of sequences which possibly overlap, and marks
them as promising. Next, the score of the alignment is computed for every promising
pair, and an arc connecting the two respective vertices from the pair is added to the
graph with the calculated score. In the second part one searches for a path in the graph
which passes through the most number of vertices. Usually, it is not possible to find
one path due to errors in the input data. Thus instead of one path several paths are
returned. In the last part of the algorithm, consensus sequences are constructed from
the paths in the graph on the base of multiple alignment procedure.
The usefulness of our algorithm has been tested on raw data coming from the
experiment on sequencing of the genome of the Prochlorococcus marinus bacteria
performed at the Joint Genome Institute. As the output of the SR-ASM algorithm one
gets small number of long contigs which are highly similar to the genome of the
bacteria. The results are compared with the outputs of other available assemblers.
10:45-12:00 Parallel sessions:
Track 1: Graphs:
1. Dietmar Cieslik, "The Steiner ratio"
Steiner's Problem is the "Problem of shortest connectivity", that means, given a finite
set of points in a metric space X, search for a network interconnecting these points
with minimal length. This shortest network must be a tree and is called a Steiner
Minimal Tree (SMT). It may contain vertices different from the points which are to be
connected. Such points are called Steiner points. If we do not allow Steiner points,
that means, we only connect certain pairs of the given points, we get a tree which is
called a Minimum Spanning Tree (MST). Steiner's Problem is very hard as well in
combinatorial as in computational sense, but, on the other hand, the determination of
an MST is simple. Consequently, we are interested in the greatest lower bound for the
ratio between the lengths of these trees, which is called the Steiner ratio (of the space
X). We look for estimates and exact values for the Steiner ratio in several metric
spaces, given old and new results.
2. Martin Charles Golumbic and Robert E. Jamison, "Tolerance and RankTolerance in Graphs: Mathematical and Algorithmic Problems"
In this talk, we set out a plan for future investigation of two global themes which
focus on special families and properties of graphs.
These involve the notion of measured intersection known as tolerance and a more
general framework called rank-tolerance. Tolerance and rank-tolerance graphs were
originally motivated by work spawned by the classical family of interval intersection
graphs.
In a rank-tolerance representation of a graph, each vertex is assigned two parameters:
a rank, which represents the size of that vertex, and a tolerance which represents an
allowed extent of conflict with other vertices. Two vertices are adjacent if and only if
their joint rank exceeds (or equals) their joint tolerance. By varying the coupling
functions used to obtain the joint rank or joint tolerance, a variety of graph classes
arise, many of which have interesting structure.
Applications, algorithms and their complexity have driven much of the research work
in these areas. The research problems posed here provide challenges to those
interested in structured families of graphs.
These include trying to characterize the class SP of sum-product graphs, where the
tolerance coupling function is the sum} of the two tolerances and the rank coupling
function is the product of the two ranks, and investigating the class of mix graphs,
which are obtained by interpolating between the min and max functions.
3. Peter Adams, Moshe Rosenfeld and Hoa Vu Dinh, "Spanning graph designs"
Classical balanced incomplete block designs (BIBD) explore decompositions of the
edges of the complete graph K_n into "blocks" (complete subgraphs) of prescribed
size(s) so that every edge of K_n appears in the same number of blocks. A natural
generalization of these designs is to extend "blocks" to other graphs. For example,
decompositions of K_n into cycles has been studied by many authors. But the most
general case, the Oberwolfach Problem is still open.
Of particular interest are spanning graph designs where the blocks are graph on n
vertices. We study two groups of spanning graph designs: spanning cubic graph
designs and spanning designs inspired by equiangular lines in R^d.
For a cubic graph G of order n to decompose K_n it is necessary that n = 6k+4. We
show various constructions of such decompositions and conjecture that for n 'large
enough" all cubic graphs of order 6n + 4 decompose k-{6n+4}.
The second group of spanning graph designs involve G(2n, n-1). These are graphs of
order 2n regular of degree n - 1. The implied spanning graph design will yield 2n-1
copies of G so that every edge of K_{2n} will appear in exactly n-1 copies of G. For
instance, one can construct 11 labeled copies of the Icosahedron (a G(12,5)) so that
every edge of K_12 will appear in exactly 5 copies. These designs turn out to be
related to Hadamard matrices, conference matrices and other interesting combinatorial
designs.
Track 2: Heuristics 2:
1. Michal Penn and Eli Troupiansky, " A heuristic algorithm for solving the
heterogeneous open vehicle routing problem"
Given a capacitated directed graph with a depot vertex and demand vertices the VRP
(Vehicle Routing Problem) is aimed at minimizing the cost of supplying the required
demand. That is, minimizing the number of capacitated vehicles needed and the cost
of the tour. The problem is known to be NP-hard and vast of the research effort is
aimed at various heuristic methods for solving the problem.
We consider a variant of the VRP, termed Open VRP (OVRP), where the
vehicles are not required to return to the depotss at the end of their tours. If, in
addition, the vehicles are of different capacities and costs, the problem is denoted as
Heterogeneous OVRP.
We present a greedy heuristic for solving the Heterogeneous OVRP. Our
greedy algorithm consists of two stages – construction of an initial feasible solution
and post-optimization for improving the initial solution. We compare our heuristic to
some known algorithms for the homogeneous case and to Cross-Entropy (CE) in the
general heterogeneous case. Our preliminary results show that the greedy heuristic out
performs CE.
2. Dorit Ron, Ilya Safro and Achi Brandt, "Fast multilevel algorithms for linear and
planar ordering problems"
The purpose of linear ordering problems is to minimize some functional over all
possible permutations. These problems are widely used and studied in many practical
and theoretical applications. We have developed linear-time multilevel solvers for a
variety of such graph problems, e.g., the linear arrangement problem, the bandwidth
problem and more. In a multilevel framework, a hierarchy of decreasing size graphs is
constructed. Starting from the given graph G0, create by coarsening the sequence
G1,...,Gk, then solve the coarsest level directly, and finally uncoarsen (interpolate) the
solution back to G0. The experimental results for such problems turned out to be
better than every known result in almost all cases, while the short running time of the
algorithms enables applications on very large graphs.
Next we have considered the generalization of such problems to two dimensions. In
many theoretical and industrial fields, this class of problems is often addressed and
actually poses a computational bottleneck, e.g., graph visualization, facility location
problem, VLSI layout, etc. In particular, we have tried to minimize various
functionals (discrete or continuous) applied to the two-dimensional objects
(associated with the graph nodes) at hand, while maintaining equi-density demand
over the planar domain. We have concentrated on answering the following three
issues: (a) minimize the total length of the connections between these objects (b)
minimize the overlap between the objects and (c) the two-dimensional space should
be well utilized.
We present a multilevel solver for a model that describes the core part of those
applications, namely, the problem of minimizing a quadratic energy functional under
planar constraints that bound the allowed amount of material (total areas of objects) in
various subdomains of the entire domain under consideration. Given an initial
arrangement for a particular graph Gi in the hierarchy, rearrange the objects under
consideration into a more evenly distributed state over the entire defined domain. This
process is done by introducing a sequence of finer and finer grids over the domain and
demanding at each scale equi-density, that is, meeting equality or inequality
constraints at each grid square, stating how much material it may (at most) contain.
Since many variables are involved and since the needed updates may be large, we
introduce a new set of displacement variables attached to the introduced grid points,
which enables collective moves of many original variables at a time, at different
scales including large displacements. The use of such multiscale moves has two main
purposes: to enable processing in various scales and to efficiently solve the (large)
system of equations of energy minimization under equi-density demands. The system
of equations of the finer scales, when many unknowns are involved, is solved by a
combination of well-known multigrid techniques. The entire algorithm solves the
nonlinear minimization problem by applying successive steps of corrections, each
using a linearized system of equations.
We demonstrate the performance of our solver on some instances of the graph
visualization problem showing efficient usage of the given domain, and on a set of
simple (artificial) placement (VLSI) examples.
3. L.F. Escudero and S. Muñoz, "A greedy procedure for solving the line design
problem for a rapid transit network"
Given a set of potential station locations and a set of potential links between them, the
well-known extended rapid transit network design problem basically consists in
selecting which stations and links to construct without exceeding the available budget,
and determining an upper bounded number of noncircular lines from them, to
maximise the total expected number of users.
A modification of this problem has been proposed in the literature to allow the
definition of circular lines provided that whichever two locations are linked by one
line at most, and a two-stage approach has also been presented for solving this new
problem. The model considered in the first stage makes possible to select the stations
and links to be constructed without exceeding the available budget, in such a way that
the total expected number of users is maximised. Once they have been selected, in the
second stage each one of these links is assigned to a unique line, so that the number of
lines passing through each selected station is minimised.
In this work we introduce an improvement in the model considered in the first stage
above to obtain a connected rapid transit network. We also present a modification of
the algorithm proposed for solving the line design problem of the second stage above;
it is a greedy heuristic procedure that attempts to minimise the number of transfers
that should be done by the users to arrive at their destinations, without increasing the
number of lines that pass through each station.
The computational experiments that will be reported will show that this greedy
procedure can significatively reduce the total number of transfers required for the
solutions obtained by the approach taken from the literature.
12:10-13:10 plenary talk – Julian Molina "Metaheuristics for multi-objective
combinatorial optimization"
Meta-heuristic methods are widely used to solve single-objective combinatorial
optimization problems, especially when dealing with real applications. However, a
multiple criteria approach is usually demanded when solving real-world applications
requiring the consideration of several conflicting points of view corresponding to
multiple objectives.
As it happens within the field of single-objective combinatorial optimization, metaheuristics have shown to be efficient for solving multi-objective optimization models.
It is one of the most active and growing branching in the field of multi-objective
optimisation in current years.
This talk provides an overview of the research in this field and visits its principal
aspects, including main methods, performance measures, test functions, and
preference inclusion techniques. New trends in the field will be outlined, including
new hybrid procedures and links to exact techniques.