COMPUTER TECHNOLOGY INSTITUTE
JANUARY 1999
TECHNICAL REPORT No. TR 99/01/01
Fast parallel approximations to extended Positive Linear
Programs applied to a new variation of matching
Pavlos S. Efraimidis and Paul G. Spirakis
January 1999
Abstract (Short)
In this paper extensions to Positive Linear Programs (PLP) are
defined, and their parallel approximation discussed. It is shown that
certain PLP extensions cannot be approximated in NC, and a new
methodology, PEPS, for fast parallel approximations to ε-relaxed PLP
extensions that admit a bounded degree violation of specific problem
constraints, is presented.
Of independent interest is the algorithm that solves maximum
matching in bipartite graphs (MMBG) in NC given a Fully NC
Approximation Scheme (FNCAS) for PLP.
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Fast parallel approximations to extended Positive Linear
Programs applied to a new variation of matching.*
Pavlos S. Efraimidis§,†,‡
Paul G. Spirakis†,‡
Abstract
In this paper extensions to Positive Linear Programs (PLP) are defined, and their
parallel approximation discussed. It is shown that certain PLP extensions cannot be
approximated in NC, and a new methodology, PEPS, for fast parallel approximations to εrelaxed PLP extensions that admit a bounded degree violation of specific problem
constraints, is presented. To the authors knowledge, this is the first time the PLP model is
extended in a general way while permitting an NCAS.
For k-extended PLP (k-ePLP) with polylog number of equality constraints PEPS
finds in NC a fractional ε-relaxed (1+ε)-approximate solution. If certain conditions hold then
this result can be extended to general k-ePLP’s that support equality constraints, covering
constraints and negative variable coefficients. PEPS uses as a subroutine the algorithm of
Luby and Nisan [LN93] for solving a series of standard PLP problems. PEPS is demonstrated
on eMWMBG, a new variation of maximum weight matching on bipartite graphs that extends
standard MWMBG with the feature to force a polylog number of nodes to be matched.
Furthermore, the remark in [TX98] that PLP which admits a linear number of
equality constraints cannot be approximated in NC within any constant factor is reinforced by
showing that the same hardness result holds of PLP with even one equality or covering
constraint. This indicates that the results of PEPS for the PLP extensions with equality or
covering constraints cannot be significantly improved.
Finally, of independent interest is the algorithm that solves maximum matching in
bipartite graphs (MMBG) in NC given a Fully NC Approximation Scheme (FNCAS) for PLP.
*
This work was partially supported by ESPRIT LTR ALCOM-IT (contract No. 20244).
The work of P.S. Efraimidis was supported in part by the Bodosaki Foundation under a
grant for PhD studies. Bodosaki Foundation, Leoforos Amalias 20, 10557 Athina, Greece.
†
Computer Engineering and Informatics Department, Patras University, 26500 Rion, Patras,
Greece.
‡
Computer Technology Institute, Kolokotroni 3, 26221 Patras, Greece. E-mail:
{efraimid,spirakis}@cti.gr
§
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1
JANUARY 1999
Introduction
Positive Linear Programming (PLP) is the subclass of Linear Programming (LP) in
packing or covering form where all coefficients of matrix A and vectors b and c are
nonnegative. In their seminal work [LN93] Luby and Nisan presented an NC Approximation
Scheme (NCAS) for PLP. Recently, Bartal et al. in [BBR97] provided a modified version of
the algorithm. These NC approximation schemes for PLP provide a powerful general tool for
parallel approximations since they can be used to approximate in NC any combinatorial
problem that can be formulated as a PLP. Two classic optimisation problems that have been
approximated by this approach are maximum matching in bipartite graphs and minimum set
cover.
PLP is a restricted subclass of LP and hence it is not strong enough to directly model all
the combinatorial optimisation problems. Despite the fact that in most combinatorial problems
it is not possible to avoid directly non-PLP features in their linear programming formulation,
for a number of them it is possible to transform the linear program into a PLP that is in a
sense equivalent to the original problem formulation. Cost can be a constant factor decrease in
the approximation ratio of the initial linear program that can be traded against a constant
factor increase in the running time. This approach has been used in [Tr96] and [Tr98] for NC
approximations to MAX-SAT, MAX DIRECTED CUT and MAX-kCSP. The linear
programs in [Tr96] and [Tr98] have a very special structure which is exploited to achieve the
NC approximation.
In this paper, extensions to Positive Linear Programs are defined and their parallel
approximation discussed. The authors show that certain direct extensions of PLP cannot be
approximated in NC, and present a methodology for fast parallel approximations to their εrelaxed versions. To the authors knowledge this is the first time that PLP is extended in a
general way while permitting an NCAS. This new model is not a generalisation of the model
in [Tr96] and [Tr98] but provides a different and much more general extension to PLP at the
cost of an increase in the running time of the algorithm.
Matching is a classical problem in the study of algorithms. It has intimate connections to
several other fundamental problems and is used as a subroutine for solving some of them.
Intensive research about its parallel complexity has been very fruitful but unanswered
questions still remain. For bipartite graphs it is known that maximum matching is in NCAS
and in RNC but it is not known to be in NC. In [LN93] the NCAS for PLP is used to
approximate the optimal value of maximum matching in bipartite graphs.
In this work a new variation of matching weight matching on bipartite graphs
(MWMBG) is defined; the extended MWMBG or eMWMBG, that adds to the standard
MWMBG problem the feature to force a number of nodes to be matched. eMWMBG is
modelled as an extended Positive Linear Program (k-ePLP), i.e. a PLP that admits a number
of non-PLP features - in this case k equality constraints. If the number of equality constraints
is small, a constant or up to polylog(n), the methodology PEPS provides an NC algorithm for
finding for any ε>0: ε=O(polylog(n)) an ε-relaxed (1+ε)-fractional approximate solution to the
problem. This result matches the best known approximation ratio for a fractional solution for
standard MWMBG in NC. The ε-relaxed solution to eMWMBG guarantees that the forced
nodes are covered by at least 1-ε. No other algorithm for dealing in NC with eMWMBG is
known to the authors. The techniques of [Co92] that are used in [DSST97] for obtaining an
integer solution to MWMBG from a fractional one can also be used for eMWMBG. However,
it is not known to what extent the produced integer solution will satisfy the forcing constraints
and so the problem of obtaining an integer solution for eMWMBG from a fractional one
remains open.
The methodology PEPS for solving the k-ePLP uses as a subroutine the algorithm of
Luby and Nisan [LN93] for solving a series of standard PLP problems. The methodology is
generalised to handle general k-ePLP problems with a limited number of equality constraints,
variables in covering constraints, and variables with negative coefficients. If certain
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conditions hold then it is shown that an ε-relaxed (1+ε)-approximate solution to the general kePLP is found in NC for any ε>0: ε=O(polylog(n)).
The non-approximability of linear programming in NC was first shown by Serna in
[Se91]. Megiddo provided an alternative proof of the same result in [Me92]. Recently,
Trevisan and Xhafa showed in [TX98] that PLP is P-Complete and that the extension of PLP
that admits equality constraints cannot be approximated in NC within any constant factor.
This remark is reinforced in the present work by showing that direct extensions of PLP with
even one equality or covering constraint cannot be approximated in NC. This result indicates
that the approximation results of PEPS for k-ePLP with equality or covering constraints
cannot be significantly improved. Figure 1-1 presents known results about the complexity of
Linear Programming and the subclasses of LP discussed in this work. The results of this work
are indicated by an asterisk.
Figure 1-1: Complexity results about Linear Programming and Subclasses
Linear Programming
Positive Linear Programming
PLP with many equality constraints
PLP with 1 equality constraint
PLP with 1 covering constraint
Exact Solution
P-Complete
P-Complete
P-Complete
P-Complete
P-Complete
Approximation
P-Complete within any constant
NCAS
P-Complete within any constant
P-Complete within any constant *
P-Complete within any constant *
The key notion for deriving NC algorithms for the k-ePLP model even though most of its
cases are proven to be non-approximable in NC, is the ε-relaxation of certain constraints of
the problem. The ε-relaxation of specific “hard” constraints permits PEPS to achieve
approximate or even superoptimal solutions by introducing a bounded degree of infeasibility
for the solution. The notion of approximations by superoptimal but infeasible solutions has
been used for sequential algorithms in [HS87] and [PST95]. Hochbaum and Shmoys in
[HS87] present approximation algorithms that aim to find superoptimal but infeasible
solutions and the performance is measured by the degree of infeasibility allowed. A relaxed
decision procedure that either finds an ε-approximate solution to a decision/search problem or
concludes that no exact solution exists is presented in [PST95].
Trevisan and Xhafa raised in [TX98] the question about the existence of a Fully NC
Approximation Scheme (FNCAS) for PLP where the authors showed that the model for CVP
they used for proving the P-Completeness of PLP cannot provide the same answer to the
FNCAS problem. A simple boolean circuit of linear depth is their counterexample. In this
work, a result of independent interest that connects two open problems in parallel complexity,
the existence of a Fully NC Approximation Scheme (FNCAS) for PLP, and an NC algorithm
for maximum matching in bipartite graphs (MMBG), is proved. An algorithm is presented
that given an FNCAS for PLP solves MMBG in NC.
1.1
Outline
Section 2 includes preliminaries, definitions and notation. In Section 3 the algorithm
PEPS for k-ePLP is presented, analysed and applied on eMWMBG. In Section 4 PEPS is
applied on general k-ePLP problems. Section 5 contains the proof that PLP with even one
equality or covering constraint cannot be approximated in NC. Section 6 presents an NC
algorithm for MMBG given a FNCAS for PLP. Finally, a discussion about the results of this
work, open problems and planned research in this field is presented in Section 7.
2
Preliminaries
An NC Approximation Scheme (NCAS) is a family of algorithms that, given a problem
instance of size n and a constant ε>0, finds a (1+ε) approximate solution in time polylog n by
using a polynomial in n number of processors. If additionally the running time is at most
polynomial in log(1/ε) and the number of processors is polynomial on 1/ε the scheme is a
Fully NC Approximation Scheme (FNCAS).
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A bipartite graph is a graph whose set of vertices can be partitioned into two subsets
V1 and V2 such that every edge of the graph joins V1 with V2. In Maximum Matching on
bipartite graphs (MMBG), a bipartite graph G=(V1,V2,E) is given and the objective is to
find the largest subset E’ of E such that each pair of edges in E’ has disjoint endpoints.
Maximum Weight Matching in Bipartite Graphs (MWMBG) is the generalisation of
MWMBG where each edge in E has a real weight and the objective is to find maximum
weight subset E’ of edges such that each pair of edges in E’ has disjoint endpoints.
Definition 2-1. Extended Maximum Weight Matching in Bipartite Graphs (eMWMBG).
The MWMBG where additionally a limited number of graph nodes can be forced to be
matched.
Definition 2-2. A Positive Linear Program (PLP) ([LN93]) is a linear program in packing
or covering form were all the coefficients of the matrix A and the vectors b and c are nonnegative.
Figure 2-1: The Packing and the Covering form of PLP
PLP in Packing form
PLP in Covering form
max cTx
min yTb
subject to:
subject to: yTAc
Axb
x0
y0
A,b,c non-negative
Theorem 2-1 ([LN93]). There is a NC algorithm that, given a Positive Linear Program in
packing or covering form and a constant ε > 0, approximates both the primal and the dual
problem within a factor 1ε. The algorithm runs in time polylog in logN and 1/ε, where N is
the number of non-zero entries of the matrix A.
The modified NCAS for PLP in [BBR97] achieves a slightly better running time.
Definition 2-3. A PLP-Violation of a k-ePLP in Packing form1 is either an equality
constraint or a variable that appears
 in one or more covering constraints with positive coefficient and/or,
 in one or more packing or equality constraints with negative coefficients.
Definition 2-4. k-ePLP. Given k, a constant or up to polylog(n), a k-ePLP is a PLP that can
have up to k PLP-Violations.
Definition 2-5. ε-Relaxed constraints. An ε-relaxed equality constraint is an equality
constraint that has been relaxed to a pair of one packing and one covering constraint that
allow a bounded relative gap ε between them. An ε-relaxed packing (covering) constraint is
a packing (covering) constraint where it is permitted to violate the right-hand side by a
bounded ratio ε.
Definition 2-6. An ε-relaxed solution to a k-ePLP is a solution that is feasible when specific
problem of the k-ePLP are ε-relaxed. An ε-Relaxed (1+ε)-approximate solution to a kFigure 2-2 Constraints and their corresponding ε-relaxations
Constraint
a
i, j
x j  bi

j
a
j
i, j
x j  bi
j
a
ε-Relaxed Constraint
ai , j x j  bi and
ai , j x j  bi (1   )

a
j
i, j
x j  bi (1   )
i, j
x j  bi (1   )
j
i, j
x j  bi
j
a
j
ePLP is an ε-relaxed solution of objective value at least 1-ε times the optimum of the exact kePLP. An ε-Relaxed Decision Procedure for a k-ePLP is an algorithm that given an
1
It is straightforward how to define a PLP-Violation for a k-ePLP in covering form.
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instance of a k-ePLP and a constant ε>0 either returns a ε-relaxed (1+ε)-approximate solution
to the k-ePLP or concludes that no exact solution exists.
2.1
Notation
In this paper the term PLP indicates a PLP in packing form unless it is explicitly
specified that it is in covering form. All the matrix elements, coefficients and the variables
that appear in the linear programs will be non-negative unless otherwise specified. A PLP has
n variables, m constraints and N on non-zero elements in the matrix. Variable names that
represent vectors are either in bold or marked with an arrow on top of them.
3
k-ePLP with k equality constraints
3.1
The eMWMBG problem formulated as a k-ePLP
Figure 3-1 presents the linear program formulation of eMWMBG. The linear program is
a PLP in packing with an additional set of equality constraints. The constraints of the linear
program guarantee that each node v is matched at most once and that all the nodes in V e must
be matched.
Figure 3-1: eMWMBG formulated as a k-ePLP
max  we xe
eE
subject to:
x
e:vej
x
e:ve
e
 1,  v  V  Ve
Packing constraints
e
 1,  v  Ve , | Ve | k
Equality constraints
xe  0, eE
The extension of PLP that admits up to k equality constraints is the model k-ePLP with
equality constraints described in Figure 3-2.If the cardinality of Ve is polylog on n than the kePLP model can be used to approximate it. In order to solve the k-ePLP a parameterised PLP
is constructed. A series of these PLP’s, that are actually ε-relaxed decision procedures for the
k-ePLP, are solved. Each PLP is approximated with an NCAS algorithm for PLP.
3.2
Algorithm PEPS
Figure 3-2: The k-ePLP model with equality constraints
Max cTx
Subject to
Ap x
 bp
Ae x
= be
x0
Given a k-ePLP with ke equality constraints first the appropriate PLP is constructed. In
the PLP all the equality constraints of the original k-ePLP problem are replaced by packing
constraints. For each equality constraint that is replaced a lagrangean-like term is introduced
into the objective function of the PLP. This term will force the packing constraint that
corresponds to the equality constraint to become tight or almost tight in the solution of the
PLP problem. The weights of the lagrangean-like terms are calculated relatively to Z.
Moreover a constraint that bounds the objective value of the original k-ePLP by Z is
introduced into the PLP.
The linear program PLP1 in Figure 3-4 is a valid Positive Linear Program and can be
solved by the NCAS algorithm of Luby and Nisan. Parameter Z corresponds to the optimal
value of the objective function of the original k-ePLP problem. For every specific value of Z,
PLP1 corresponds to a search problem that is solved with Luby & Nisan’s PLP algorithm as a
ε-relaxed decision procedure. Given PLP1 and a value Z the NCAS of Luby and Nisan finds a
vector x satisfying the constraints in Figure 3-5. This vector x is a (1+ε)-approximate solution
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Figure 3-3: Algorithm PEPS for k-ePLP
Input:
Output:
Pseudocode:
A k-ePLP instance and a constant ε>0
Either an ε-relaxed (1+ε)-approximate solution to the k-ePLP
or the k-ePLP is infeasible.
[1]
Construct the appropriate PLP
[2]
Choose lower bound L and upper bound U
[3]
Loop: do log((U+L)/(Lε)) times.
[3.1]
Z = ( L + U) / 2
[3.2]
approximate PLP(Z) with the NCAS
[3.3]
feasible ?
[3.3.1]
No: U = Z.
[3.3.2]
Yes: L = Z.
[4]
End Loop
to the PLP of Figure 3-4 and a ε-relaxed (1+ε)-approximate solution to the original k-ePLP in
Figure 3-2.
Algorithm PEPS finds an approximate maximum value of the objective function of the
packing problem by searching for the maximum value of Z for which the ε-relaxed decision
procedure on PLP1 returns an (1+ε)-approximate solution. The search is done within a binary
search procedure that either returns a ε-relaxed (1+ε)-approximate solution to the k-ePLP or
decides that there is no exact solution. Given a lower bound L and an upper bound U on the
optimal objective value of the k-ePLP and a constant ε>0 the algorithm needs log((U-L)/ε)
steps for finding an approximate solution of absolute error ε>0 and log((U-L)/(Lε)) for an
error ratio of ε. At each step of the binary search an ε-relaxed decision procedure is solved.
For eMWMBG the term log((U-L)/(Lε)) is O(log(n/ε)). If the heaviest weight of the
bipartite graph is W then trivial lower and upper bounds for the eMWMBG problem are W
Figure 3-4: PLP1 - The PLP that will be solved
max cTx
subject to
+ we Ae x
Ap x
Ae x
cTx
 bp
 be
 Z
wei 
x  0,
z1
kebei
and nW respectively. So the application of the algorithm on eMWMBG gives an algorithm
that runs in time polynomial in logN, 1/ε and ke and returns a (1+ε)-approximate fractional
solution where the k forced nodes are covered by at least 1-ε.
3.3
Intuition of Algorithm PEPS
Figure 3-5: Relaxed Approximate Solution x
be(1-ε’)
Z(1-ε’)


Ap x
Ae x
cTx
x




bp
be
Z
0
Algorithm PEPS combines the different versions of a problem, the decision version, the
search version and the optimisation version. In a decision problem it is asked to determine
whether a certain object exists. In a search problem, on the other hand, it is required to
produce the object when it exists. In an optimisation problem it is asked to find among certain
objects the one of minimum or maximum cost.
Positive Linear Programming is a method to solve optimisation problems. In the present
approach the optimisation feature of PLP is used for forcing certain relaxed constraints to
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become tight. For this reason instead of solving a real optimisation problem the PLP is
actually used to solve a search version of the k-ePLP problem, that decides is a certain
objective value is feasible and if yes it returns a corresponding solution vector. Given positive
upper and lower bounds on the objective value of the k-ePLP, the constant ε>0 solving a
logarithmic on the problem size and 1/ε number of search problems, is enough to find a (1+ε)approximate solution to the k-ePLP.
3.4
Approximation Ratio
In the analysis, for simplicitly, the slightly inaccurate equalities (1+ε 1)(1+ε2)=(1+ε1+ε2)
and (1/(1+ε))=(1-ε) will be used. This is done only a constant number of times. Given the
constant ε>0 for the approximation ratio εp is defined to be εp=ε/2ke. First certain properties
of the given k-ePLP, the value Z and the corresponding PLP are shown.
Claim 3-1: Given Z, every feasible solution to the PLP has objective value equal or less to
2Z.
Proof: The objective function of the PLP is (cTx + weAex). The first term is upper bounded by
Z: cTx  Z. The same is true for the second term:
we Ae x   wei Aei x  wei bei 
i
i
i
Z i
be  Z
k e bei
■
Claim 3-2: A feasible solution to the PLP of objective value 2Z achieves the upper bound of
all the terms in the objective function.
Proof: The analytical form of the objective function is:
c T x  we Ae x  c T x  we1 Ae1 x  ..  weke Aeke x
The bounds for each of the terms in the objective function are cTx  Z and
i  k e : wei Aei x  wei bei . If any of these inequalities is strictly satisfied then the objective
value is strictly less than 2Z.
■
Claim 3-3: If there is a feasible solution to the PLP of objective value 2Z then this solution is
optimal.
Proof: From claim 3-1.
■
Claim 3-4: If there is a feasible solution to the k-ePLP of objective value Z then the optimal
value of the corresponding PLP for the value Z is 2Z.
Proof: It is enough to show that the feasible solution of the k-ePLP corresponds to a feasible
solution of the PLP with objective value 2Z.
Feasibility: Let x be a feasible solution to the (exact) k-ePLP problem in Figure 3-2 and with
objective value Z. Then the solution x satisfies all the constraints of the PLP since
Ap x
bp

Ae x =
be
cTx
=
Z
and so x corresponds to a feasible solution for the PLP. The objective value of the feasible
solution is : (cTx + weAex) = Z + Z = 2Z.
■
Theorem 3-1. If there is a feasible solution to the k-ePLP of objective value Z then the (1+ε)approximation to PLP1 with parameter Z returns an ε-relaxed (1+ε)-approximate solution to
the k-ePLP.
Proof: Assume that the value Z is feasible for the k-ePLP. Then by Claim 3-4 the optimal
value to the PLP is 2Z. Solving the PLP with the NCAS of [LN93] returns a feasible (1+ε)approximate solution x to the PLP. Let θ be the absolute error of the objective value of the
approximate solution, then: 2Z(1-ε)  cTx + weAex  2Z  0  θ  2Zε . The max deviation
for each term in the objective function cannot exceed the total error in the objective value:
(*)
Z-θ  cTx  Z
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Z – θ  weAex  Z
(**)
Objective Value:
(*)  Z - 2Zε  cTx  cTx  Z (1 - 2ε)
If Z is a (1+ε)-approximation of the optimal value of the k-ePLP then:
cTx  OPT(1-ε)  Z(1-ε)(1-2ε)  cTx  OPT (1-ε-2ε)
ε-Relaxed equality constraints:
(**)
 i, wei bei    w A x  wei bei  i, w A x  wei bei  2Z
 i,
Z 
Z i
A x 
b  2Z  i, A x  bei  2k e bei 
i
i e
k e be
k e be
■
 i, A x  bei (1  2k e )
Corollary 3-1. If the (1+ε)-approximation to PLP1 for Z does not return an ε-relaxed (1+ε)approximate solution to the k-ePLP then there is no feasible solution to the k-ePLP of
objective value Z.
Note 3-1. The Theorem and the Corollary show that the algorithm is an ε-relaxed decision
procedure for the k-ePLP and a specific value Z.
Let L be a lower bound on the optimal value of the k-ePLP and U an upper bound and δ>0 be
the lowest bound such that OPT-Ζδ.
Lemma 3-1. The value Z returned by the binary search procedure is a (1+ε)-approximation2
of the optimal value of the k-ePLP. The binary search needs O(log((U-L)/(Lε))) steps for L
strictly positive.
Proof: At step 1 of the binary search: δ  (U-L)/2 and at step k: δ  (U-L)/2k.
We can now estimate the minimum large enough integer value of s for bounding the absolute
error εα and the relative error ε.
Absolute error:
   
  U  L 
U  L 
U L

  s  log 
 s  log 
k


2




  
The lower bound L has to be strictly positive for achieving a relative error bound.
Relative error:
  
U L
  U  L 
U  L 
 s r  log 
  s r  log 

k
L2
 L 
  L 
Depending on the specific application the value of k can depend only on the instance size
and the approximation ratio. For example for MWM on bipartite graphs with maximum edge
weight W trivial bounds are L=W and U=n*W. The corresponding sr is sr=O(log(n/ε)).
Theorem 3-2. Algorithm PEPS returns an ε-relaxed (1+ε)-approximate solution to the kePLP.
■
Proof: From Theorem 3-1 and Lemma 3-1.
3.5
Complexity

 
 U  L 
log n log m /  p
Theorem 3-3: Algorithm PEPS runs in time  log 
 log( nm)
  L 
p 4

O(N) processors.



on
2
By approximation it is meant that ZOPT(1-ε). No upper bound on Z relative to OPT is
shown. This issue is discussed in Section 3.6.
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Proof: The binary search procedure needs O(log((U – L) / (Lε))) steps. At each step of the
binary search the NCAS algorithm of Luby & Nisan is used to solve a PLP. For a PLP with n
variables and m constraints the NCAS runs in time O(log(nm) log(n) log(m/ε p) / (εp)4). The
most processor demanding operation of the algorithm is the NCAS for PLP of Luby and
Nisan that runs on O(N) processors, where N is the number on non-zero elements of the
matrix. The other operations of the algorithm run on less processors. It is important to note
that since the NCAS is used to solve PLP and not the k-ePLP the actual size of the problem is
the size of the PLP. However the difference is within a constant factor and hence it can be
assumed that the parameters n,m and N refer to the k-ePLP.
The running time can be improved by a factor of 1/εp if the NCAS for PLP of [BBR97] is
used instead of Luby and Nisan’s algorithm.
■
3.6
Comments on PEPS
There are several important notes on algorithm PEPS.
 ε-Relaxed Approximate Solutions. If algorithm PEPS finds no solution then the exact
problem is not feasible. If PEPS finds a solution x then only information on OPT is that its
upper bound is (1+ε)cTx. There is no lower bound on OPT and there is no guarantee that the
exact problem is even feasible. If there would be an NC alogrithm to lower bound OPT within
a constant factor then by combining it with algorithm PEPS the value of OPT would be
estimated within a constant factor. This would give an approximation to the exact k-ePLP and
that is a contradiction because of Theorem 5-1 and Corollary 5-1. In this sense the result
found by PEPS cannot be signifanctly improved. The same question for k-ePLP with negative
variable coefficients remains unanswered.
 Feasibility. The problem of feasibility for standard PLP’s is a trivial issue while for
extended problems like k-ePLP not all instances are feasible. Testing the ε-relaxed feasibility
of the k-ePLP can be done by solving a ε-relaxed decision procedure with objective function
that contains only lagrangean-like terms. The appropriate PLP is shown in Figure 3-7 and the
algorithm in Figure 3-8.
Figure 3-7: PLP for checking the (ε-relaxed) feasibility of the k-ePLP.
Max
Subject to
we Ae x
Ap x
Ae x
x  0,


wei 
bp
be
1
k e bei
Figure 3-8: Algorithm for checking feasibility of a k-ePLP
Input:
Output:
A k-ePLP instance
A constant ε>0
Either a relaxed (1+ε)-approximate solution to the k-ePLP
or the k-ePLP is infeasible.
Pseudocode:
[1]
Construct the appropriate PLP.
[2]
Search for a relaxed (1+ε)-approximate solution to the PLP.
[2]
Solution found?
[2.1]
No: The k-ePLP is infeasible.
[2.2]
Yes: There is a feasible solution to the relaxed k-ePLP.
 Bounds. An important issue for the algorithm is the specification of the bounds on the
optimal value of the k-ePLP. The bounds are necessary for the binary search procedure and
their value is very important since it depends on them if the algorithm will be in NC. A simple
problem independent procedure for finding the bounds is presented. However it cannot be
guaranteed that the procedure will always run in polylog time. The first step is be to solve the
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JANUARY 1999
feasibility-checking problem of 3.7.2. If the problem is feasible then the objective value of the
solution returned by the algorithm can be used as a lower bound. The upper bound can be
found by doubling. It might be possible that the binary search procedure of PEPS cannot
decide at its first step which half to choose. If the first middle value of the bounds is not
feasible then the binary search would not be able to decide in which half the solution is. The
feasibility-checking problem of 3.7.2 can decide if there are feasible solutions and in which
half of the current range.
 Superoptimality. The ε-relaxed (1+ε)-approximate solution to the k-ePLP (Figure 3-5)
can be scaled to a superoptimal ε-relaxed solution without increasing significantly the
violation of each constraint. In this case all the packing constraints must be ε-relaxed.
4
The general k-ePLP model
In this section the general k-ePLP model that can support a constant number of any type
of PLP-Violations is defined. Techniques similar to the one for handling equality constraints
are used to handle the other PLP-violations, covering constraints and variables with one or
more negative coefficients.
The general model and the restrictions in Figure 4-1 describe a general k-ePLP model.
Let OPT be the optimal value of the objective function of the k-ePLP problem. The algorithm
finds the solution x presented in Figure 4-2. The simpler cases of the k-ePLP model can be
extracted from the general one by simply replacing the non-existing Tables with zeros. For
the general case with all types of PLP-Violations if the restrictions of Figure 4-1 hold then the
solution is an ε-relaxed (1+ε)-approximate solution to the k-ePLP (Last Column of Figure 42). A superoptimal solution can be obtained by scaling the solution of Figure 4-3.
Theorem 4-1. Given a k-ePLP problem instance and a constant ε>0, if the restrictions of
Figure 4-1 hold then algorithm PEPS finds an ε-relaxed (1+ε)-approximate solution to the
Figure 4-1: The general k-ePLP model and the additional restrictions
max cTx
subject to
x

=

Ap x - Ap,n x
Ae x - Ae,n x
Ac x - Ac,n x
x0
bp
be
bc

1 Ap,n
1Ae,n
1 Ac
D




=
1
dp bp
de be
dc bc
max ( dp, de, dc )
problem in time polynomial in logN, 1/ε, κ, d and log((U-L)/(Lε)). For given constants ε, k
and d the algorithm runs on O(N) processors and is in NC provided that log((U-L)/(Lε)) =
O(polylog(N)).
Proof: In Section 3 it has been shown how algorithm PEPS applies on k-ePLP with equality
constraints. In the following sections it is shown how PEPS handles one by one all the other
k-ePLP violations and any valid combination of k-ePLP violations. The basic approach for
handling each PLP-violation and the valid combinations of them is the same as in Section 3.
Figure 4-2: The solution before and after the restrictions of Figure 4-1

(1-2εp)OPT
Ap x - Ap,n x

Ae x - Ae,n x
Ae x - Ae,n x
Ac x - Ac,n x



 OPT

bp + 2kεp 1 Ap,n

x0
bp (1 + 2kdεp)



be (1 + 2kdεp)
be (1 – 2kdεp)
bc (1 – 2k(d+1)εp)
cTx

be + 2kεp 1 Ae,n
be – 2kεp 1 Ae,n

bc – 2kεp(bc + 1 Ac)
The difference lies in the construction of the appropriate PLP. After the PLP has been
constructed the same binary search procedure of PEPS is used to approximate the relaxed
version of the problem. The general case of Figure 1 can be handled by the integration of all
the separate cases and this proves Theorem 4-1.
■
TECHNICAL REPORT No. TR99/01/01
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COMPUTER TECHNOLOGY INSTITUTE
4.1
JANUARY 1999
Negative Coefficients
The linear program in Figure 4-3 is a k-ePLP where up to kn variables can have negative
coefficients. For each constraint the sum of the absolute values of the negative coefficients
must be bounded with respect to the right hand side.
Figure 4-3: A k-ePLP where k variables can have bounded negative coefficients
max cTx
subject to:
Ax – Anxn  bn
xn  1
In the first step the LP of Figure 4-3 is transformed to the LP of Figure 4-4 by
introducing for each of the kn “illegal” variables xi an artificial variable yi such that xi + yi = 1.
Clearly the two LP's are equivalent. In the new LP the negative coefficients have been
eliminated and instead k equality constraints have been introduced into the problem. These
Figure 4-4: Elimination of the negative coefficients
max cTx
subject to:
Ax + Anyn  b +
An
xn + yn = 1
x, yn  0
A, An, bn, c 0
Figure 4-5: PLP2 - The PLP that will be solved
max cTx + w(xn+yn)
st: Ax + Anyn  bn + An
cTx  Z
xn + yn  1
x, yn  0
A, An, bn, c 0
equality constraints are treated as in section 3-3; they are converted to packing constraints and
in order to push the new inequalities to become tight or r at least almost tight a lagrangean
like term is introduced into the objective function for each of them. Additionally a new
constraint that bounds the objective function of the original k-ePLP is introduced. The final
linear program is PLP2 (Figure 4-5), a PLP that can be approximated by an NCAS for PLP.
The weight vector w of the lagrangean terms is Z/kn*1.
value Z, an ε-relaxed decision procedure for the k-ePLP and the value Z is obtained.
Claims 3-1,3-2,3-3 and 3-4 of Section 3-5 hold.
Proof: In the same way as in Section 3-5.
■
Theorem 4-2. If there is a feasible solution to the k-ePLP of objective value Z then the (1+ε)approximation to PLP1 with parameter Z returns an ε-relaxed (1+ε)-approximate solution to
the k-ePLP.
Proof: The proof is described briefly since it is very similar to the proof of Theorem 3-1.
Assume that there is an exact feasible solution to the k-ePLP of objective value Z and the PLP
is solved with the NCAS of [LN93]. Let x be the solution found and ζ the objective value of
x. ζ1=cTx and ζ2=w(xn+yn). Clearly ζ=ζ1+ζ2. Let θ be the absolute error in the objective
function θ=2Z-ζ. Because of the assumption and claim 4: 2Z(1-ε) cTx + w(xn+yn)2Z and
because of this 0θ2Zε. The absolute error on each of the terms ζ1 and ζ2 is at most θ.
(*)
    1  Z     cT x  Z
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COMPUTER TECHNOLOGY INSTITUTE
Z    2  Z 
JANUARY 1999
Z    we Ae x  Z
(**)
Objective Value:
(*)  Z – 2Zε  cTx  cT  Z(1 - 2ε)
ε-Relaxed equality constraints:
(**)
 i, wni    wn ( x ni  y ni )  wni  i, wn ( x ni  y ni )  wni  2Z
 i,
Z i
Z
( x n  y ni ) 
 2 Z
kn
kn
 i, ( xni  yni )  1  2kn
(***)

(***)  i, yni  1  2kn  xni  i, An yn  An  An 2kn  An xn
Since x is a feasible solution to the PLP, it satisfies the packing constraints:


Ax  An yn  bn  An  Ax  An  An 2kn  An xn  bn  An  Ax  An xn  bn  An 2kn

If the negative coefficients are bounded 1 * An  d  bn then
An xn  bn (1  2k n d ) .
Given the upper bound U and the lower bound L on the objective value of the k-ePLP the
binary search procedure can be applied. We apply Algorithm PEPS (Figure 3-3) for finding a
ε-relaxed (1+ε)-approximate solution to the k-ePLP.
■
Let L and U be a lower bound and an upper bound, respectively, on the optimal value of the
k-ePLP.
Lemma 4-1. The value Z returned by the binary search procedure of algorithm PEPS is a
(1+ε)-approximation3 of the optimal value of the k-ePLP. The binary search needs O(log((UL)/(Lε))) steps for L strictly positive.
Proof: Similar to the proof of Lemma 3-1.
Theorem 4-3. Algorithm PEPS returns an ε-relaxed (1+ε)-approximate solution to the kePLP.
Proof: From Theorem 4-2 and Lemma 4-1.

 
 U  L 
log n log m /  p
Theorem 4-4: Algorithm PEPS runs in time  log 
 log( nm)
  L 
p 4

O(N) processors.
Proof: Similar to the proof of Theorem 3-3.
4.2



on
■
Covering Constraints
The linear program in Figure 4-6 is a k-ePLP where up to kc variables can participate
with positive coefficients in covering constraints. For each covering constraint the sum of the
values of the positive coefficients must be bounded with respect to the right hand side. With
Algorithm PEPS first the k-ePLP is converted to an appropriate PLP and then the binary
search procedure for the maximum Z is applied.
To construct the PLP, an artificial variable yi is introduced for each of the kc “illegal”
variables xi and the corresponding equality constraint xi + yi = 1 is applied. The new LP is
shown in Figure 4-7. Clearly this LP is equivalent to the initial k-ePLP. The introduction of
the complementary variables y transforms the covering constraints to packing constraints at
the cost of introducing kc equality constraints into the problem.
3
By approximation here it is meant that ZOPT(1-ε). There is no upper bound of Z relative to
OPT. This issue is discussed in Section 3-6.
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JANUARY 1999
The equality constraints are handled as in Section 3; they are converted to packing
constraints and for each one a lagrangean like term is introduced into the objective function.
Figure 4-6: A k-ePLP where kc variables can have positive coefficients in covering
constraints
max cTx
subject to:
Ax  b
Acxc  bc
xc  1
Figure 4-7: Elimination of the covering constraints
max cTx
subject to:
Ax + Anyn  b
Acyc  1Ac-bc
xn + yn = 1
x, yn  0
A, An, bn, c 0
Additionally a new constraint that bounds the objective function of the original k-ePLP is
introduced. The final linear program is PLP3 in Figure 4-8, a PLP that can be approximated
with NCAS for PLP. The weight vector w of the lagrangean terms is Z/kc*1.
Claims 3-1,3-2,3-3 and 3-4 of Section 3-5 hold.
Figure 4-8: PLP3 - The PLP that will be solved
max cTx + wc(xc+yc)
subject to:
Ax  b
Acyc  Ac - bc
cTx  Z
xc + yc  1
x, yc  0
A, Ac, bc, c 0
Proof: In the same way as in section 3-5.
■
Theorem 4-5. If there is a feasible solution to the k-ePLP of objective value Z then the (1+ε)approximation to PLP1 with parameter Z returns an ε-relaxed (1+ε)-approximate solution to
the k-ePLP.
Proof: The proof is very similar to the proof of Theorem 3-2. It is assumes that there is an
exact feasible solution to the k-ePLP of objective value Z. The PLP is solved with the NCAS
of [LN93]. Let x be the solution found, ζ the objective value of x, ζ1=cTx, and ζ2=wc(xc+yc).
Clearly ζ=ζ1+ζ2. Let θ be the absolute error in the objective function θ=2Z-ζ. Because of the
assumption and claim 4: 2Z(1-ε) cTx + wc(xc+yc)2Z and because of this 0θ2Zε. The
absolute error on each of the terms ζ1 and ζ2 is at most θ.
(*)
    1  Z     cT x  Z
Z    2  Z 
Z    we Ae x  Z
(**)
Objective Value:
(*)  Z – 2Zε  cTx  cTx  Z(1-2ε)
ε-Relaxed equality constraints:
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 i, wci    wc ( xci  y ci )  wci  i, wc ( xci  y ci )  wci  2Z
(**)
 i,
Z i
Z
( xc  y ci ) 
 2Z
kc
kc
 i, ( xci  yci )  1  2kc
(***)

(***)  i, y  1  2kc  x  i, Ac yc  Ac  Ac 2kc  Ac xc
Since x is a feasible solution to the PLP, it satisfies the packing constraints:
i
c
i
c


Ac yc  Ac  bc  Ac  Ac 2kc  Ac xc  Ac  bc  Ac xc  bc  Ac 2kc

If the non-zero coefficients of Ac are bounded 1 * Ac  d  bc then Ac xc  bc (1  2kc d ) . ■
Let L be a lower bound on the optimal value of the k-ePLP and U an upper bound and δ>0 be
the lowest bound such that OPT-Ζδ.
Lemma 4-2. The value Z returned by the binary search procedure of algorithm PEPS is a
(1+ε)-approximation4 of the optimal value of the k-ePLP. The binary search needs O(log((UL)/(Lε))) steps for L strictly positive.
Proof: Similar to the proof of Lemma 3-1.
Theorem 4-6. Algorithm PEPS returns an ε-relaxed (1+ε)-approximate solution to the kePLP.
Proof: From Theorem 4-2 and Lemma 4-1.

 
 U  L 
log n log m /  p
Theorem 4-7: Algorithm PEPS runs in time  log 
 log( nm)
  L 
p 4

O(N) processors.
Proof: Similar to the proof of Theorem 3-3.
4.3



on
■
Negative Coefficients in Covering Constraints
This case is handled like normal covering constraints. The variables that participate in a
covering constraint with negative coefficient will appear in the transformed problem with a
positive coefficient in the corresponding packing constraint. The PLP-violations are only
variables that appear in at least one covering constraint with a positive coefficient. Let k c be
the number of such variables.
Figure 4-9: A k-ePLP with negative coefficients in covering constraints
max cTx
subject to:
Ax  b
Acxc – An,cxn,c  bc
xc  1
Figure 4-10: Elimination of the negative coefficients and the covering constraints
max cTx
subject to:
Ax  b
Acyc + An,cxn,c  1Ac - bc
xc + yc = 1
xc  1
4
By approximation here it is meant that ZOPT(1-ε). There is no upper bound of Z relative to
OPT. This issue is discussed in Section 3-6.
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COMPUTER TECHNOLOGY INSTITUTE
JANUARY 1999
Figure 4-11: PLP6 - The PLP that will be solved
max cTx
subject to:
+ wc(xc + yc)
Ax  b
Acyc + An,cxn,c  1Ac - bc
cTx  Z
xc + yc  1
xc  1
Theorem 4-8. If there is a feasible solution to the k-ePLP of objective value Z then the (1+ε)approximation to PLP1 with parameter Z returns an ε-relaxed (1+ε)-approximate solution to
the k-ePLP.
Proof: The weight vector of the lagrangean terms is wc = (Z/kc)*1. The PLP is approximated
in NC within εp and if the Z is a feasible value then the solution satisfies:
Z(1-εp)  cTx  Z
Acyc + An,cxn,c  1Ac – bn,c
1 - 2kcεp  xc + yc  1
From these relations we get
Acxc - An,cxn,c  bc – 2kcεp1Ac
Given that 1Ac  dbc we get
Acxc - An,cxn,c  bc ( 1 – 2kcεp)
■
Because of Theorem 4-8 Algorithm PEPS can be used to find an ε-relaxed (1+ε)-approximate
solution to the k-ePLP. The proof is similar to the proofs of the other cases.
4.4
Negative Coefficients in Equality Constraints
The case of equality constraints with negative coefficients combines two types of PLPviolations and is the most complicated to handle. Such a k-ePLP is presented in Figure 4-12.
Let ke be the number of equality constraints, kn the number of variables with one or more
negative coefficients and k the total of PLP-violations k=ke+kn. First the negative coefficients
are eliminated with the introduction of complementary variables like in section 4-3 and then
the equalities are handled like in section 3. The final linear program that will be solved is
shown in Figure 4-14.
Figure 4-12: A k-ePLP with negative coefficients in equality constraints
max cTx
subject to:
Ax  b
Aexe – An,exn,e = bn,e
xe,n  1
Figure 4-13: Elimination of the negative coefficients
max cTx
subject to:
Ax  b
Aexe + An,eyn,e = bn,e+Ae,n
xe,n + ye,n = 1
xe,n  1
The weight vectors of the lagrangean terms are we: we* (bn,e+1Ae,n) =(Z/k)*1
and we,n=(Z/k)*1.
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Figure 4-14: PLP5 - The PLP that will be solved
max cTx
subject to:
+ we(Aexe + An,eyn,e) + we,n(xe,n + ye,n)
Ax  b
Aexe + An,eyn,e  bn,e+1Ae,n
cTx  Z
xe,n + ye,n  1
xe,n  1
Theorem 4-9. If there is a feasible solution to the k-ePLP of objective value Z then the (1+ε)approximation to PLP1 with parameter Z returns an ε-relaxed (1+ε)-approximate solution to
the k-ePLP.
Proof: The PLP is approximated in NC within εp and if the Z is a feasible value then the
solution satisfies:
Z(1-εp)  cTx  Z
(bn,e+1Ae,n)(1-2kεp)  Aexe + An,eyn,e  bn,e+1Ae,n
1 - 2kεp  xe,n + ye,n  1
From these relations it can be shown that
Aexe – An,exn,e  bn,e + 2kεp 1Ae,n
Aexe – An,exn,e  bn,e – 2kεp(bn,e +
1Ae,n)
Given that 1Ae,n  dbe,n we get
Aexe – An,exn,e  bn,e (1+ 2dkεp)
Aexe – An,exn,e  bn,e (1 – 2(d+1)kεp)
■
Because of Theorem 4-9 Algorithm PEPS can be used to find an ε-relaxed (1+ε)-approximate
solution to the k-ePLP. The proof is similar to the proofs of the other cases.
5
Parallel Complexity of PLP extensions
In [TX98] Trevisan and Xhafa showed that finding the optimal solution to a PLP is a PComplete problem. An additional outcome of that work was their remark that the extension of
PLP where a linear number of equality constraints is admitted is P-hard to be approximated
within any constant factor. In this work this remark is reinforced by showing that PLP where
even one equality constraint is admitted is P-hard to be approximated within any constant
factor. An obvious consequence of this result is that PLP where even one covering constraint
is admitted is also P-hard to be approximated within any constant factor.
Figure 5-1: LP1 models the OR-NOT CVP
max
Subject to:
tm
tk = 1, k  In1 and tk = 0, k  In0
tk  ti,, tk  tj, tk  ti+tj,
tk = 1 - tj
0  ti  1
 ( i, j, k)  OR
 ( j, k)  Neg
 i  { 1,…,m}
In the OR-NOT Circuit Value Problem (OR-NOT CVP) the objective is to find the
value of the output gate given an encoding of a boolean circuit with n gates of types OR and
NOT together with an input assignment. The problem is known to be P-Complete [GHR95].
Theorem 5-1. The extension of PLP where at least one equality constraint is admitted is Phard to be approximated within any constant factor.
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Proof: The proof is based on a reduction similar to the one used in [TX98]. An OR-NOT
Circuit Value Problem (OR-NOT CVP) is reduced to an NCAS for PLP with one equality
constraint. Since OR-NOT CVP is known to be P-Complete the proof follows. Given an ORNOT CVP it is modelled as the linear program LP1 in Figure 5-1. A binary variable is
introduced for every input and for every output of a gate. For every gate appropriate
constraints that relate its inputs and its output variables are defined.
Figure 5-2: LP2
max
Subject to:
tm
tk = 1, k  In1 and fk = 1,  k  In0
fk + ti  1, fk + tj  1, fi + fj + tk  2
tk + tj = 1
0  ti  1
 ( i, j, k)  OR
 ( j, k)  Neg
 i  { 1,…,m}
Claim 5-1 (TX[98]): The linear pogram LP1 correctly models the OR-NOT CVP Problem.
LP1 has only one feasible solution and that is equal to the outcome of the ORNOT boolean
circuit.
A simple conversion of LP1 to a PLP with one equality constraint is presented. First all the
negative coefficients are eliminated by introducing for each variable t i a complementary
variables fi, such that ti + fi = 1. The result is the linear program LP2 presented in Figure 5-2.
Claim 5-2 ([TX98]). The linear program LP2 has only one feasible solution and this solution
corresponds to the unique feasible solution of LP1.
In LP2 all the negative coefficients have been eliminated but additional equality
constraints have been introduced to the linear program. The next step is to convert the
equality constraints to packing constraints and add one new equality constraint to the
problem. This equality constraint contains an appropriate term for each of the original
equalities that have been modified to packing constraints. The result is the linear program
presented in Figure 5-3; a PLP extended with one equality constraint.
Figure 5-3: PLP2 – a PLP extended with one equality constraint
max
subject to:
tm
t   f
kIn1
k
kIn 0
k

 (t
( k , j )Neg
m
k
 t j )   (ti  f i ) | In1 |  | In0 |  | Neg | m
i 1
fk + ti  1, fk + tj  1, fi + fj + tk  2
tk + tj  1
ti + fi  1
0  ti, fi  1
 ( i, j, k)  OR
 ( j, k)  Neg
 i  { 1,…,m}
 i  { 1,…,m}
Claim 5-3. PLP2 has only one feasible solution and this solution corresponds to the unique
solution of LP2.
Proof: PLP2 differs from LP2 in that the equality constraints of LP2 have been relaxed to
packing constraints. The equality constraint of PLP2 forces all the packing constraints that
correspond to equalities in LP2 to become tight. So PLP2 and LP2 are equivalent.
■
Lemma 5-1. The linear program PLP2 is a PLP program extended with one equality
constraint and has only one feasible solution. The unique solution is equal to the output of the
OR-NOT CVP.
Proof: From Claims 5-1, 5-2 and 5-3.
■
This completes the proof of Theorem 5-1.
■
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Corollary 5-1. The extension of PLP in packing form where at least one covering constraint
is admitted is P-hard to be approximated within any constant factor.
6
A comment on FNCAS for PLP
The results that PLP is P-Complete and that PLP is in NCAS give an almost tight
description of the parallel complexity of PLP. However, there is still a gap between the two
results. It is not known whether PLP admits a Fully NC Approximation Scheme (FNCAS).
The question if PLP admits an FNCAS was raised by Trevisan and Xhafa in [TX98].
In this section an interesting result is shown that connects this problem to another wellknown open problem in the field of parallel complexity: Maximum matching in bipartite
graphs (MMBG). Given a FNCAS for PLP an NC algorithm for MMBG is presented. The
proof is simple and first uses the FNCAS for finding an approximate fractional solution to
MMBG of absolute error at most ½ and then algorithm round of [GPST89] to obtain an
optimal matching from it.
Figure 6-1: Maximum Cardinality bipartite matching formulated as a PLP.
max  xe
eE
x
subject to:
e:vej
e
 1,  v  V
xe  0 for e  E
Theorem 6-1. If there is a FNCAS for PLP then there is a NC-algorithm for Maximum
Cardinality Matching on Bipartite Graphs.
Proof: The maximum cardinality matching on bipartite graphs is formulated as a PLP. If the
bipartite graph has 2n nodes then the maximum cardinality matching has at most n edges and
the objective value of the corresponding PLP formulation is at most n. The approximation
ratio ε=1/(2n) is chosen. This value is acceptable since the running time of the FNCAS
depends on the log of 1/ε.
Claim 6-1. The FNCAS will return a fractional approximate matching of objective value at
most ½ less then the optimal matching.
Proof: Let x be the fractional approximate solution found by the FNCAS for ε=1/(2n). Then:
x
eE
e
 2n  12 .
 OPT (1   )  OPT   xe  OPT    n  1
eE
■
Lemma 6-1 ([GPST89]). Given a fractional approximate matching of objective value at most
½ less then the optimal matching the algorithm Round of [GPST89] finds an optimal
matching in time O(logn*lognC). C is the maximum weight of the edges.
This completes the proof of Theorem 6-1.
■
If the edges of a bipartite graph have integer weights bounded by a polynomial on n then in
the same way it can be shown that an FNCAS for PLP can solve in NC the maximum weight
matching on such bipartite graphs.
Corollary 6-1. If PLP is in FNCAS then MCMBG is in NC. If MCMBG is not in NC then
there is no FNCAS for PLP.
7
Discussion
In this work, the methodology PEPS for fast parallel approximations to extended Positive
Linear Programs was presented. It was shown that for a k-ePLP with a limited number of
PLP-violations, PEPS finds in NC an ε-relaxed (1+ε)-approximate solution to the problem.
Furthermore, PLP with even one equality or covering constraint was proven P-hard within
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any constant and hence for these two cases ε-relaxed approximations are the best possible
solution in NC. Since it is proven that the direction of permitting additional type of constraints
in a PLP leads to P-hard problems, the only direction left to extend PLP in a general way is
the permission of negative variable coefficients. It is open if and when PLP with support for
negative variable coefficients is P-hard. For all kinds of PLP-violations it is open if a larger
than polylog number of them can be admitted.
Despite the fact that the authors consider the results of this work mainly of theoretical
interest, there is a strong interest in their practical evaluation. Currently the techniques
presented in this work are implemented using the well-known LP/Integer LP environment of
“lp_solve”. It is planned to evaluate Algorithm PEPS both on top of Luby and Nisan’s
algorithm and Bartal et al.’s algorithm. If the techniques are proven efficient it could be
interesting to consider their incorporation to standard environments such as LEDA.
The new variation of matching, eMWMBG, appears as an interesting problem for
applications of matching itself, but also for cases that can be reduced to matching, such as
maximum flow [DSST97], or cases where matching is used as a sub-problem. An interesting
open question concerns the rounding of the fractional relaxed solutions of eMWMBG to a
feasible matching. Even though it is obvious that for the unweighted case the techniques of
[CO92] will yield a feasible approximate solution with the same approximation ratio it is not
clear if it will fulfil the constraints of forced nodes. Forced nodes are covered by at least 1-ε
and hence one could expect that they will be matched with high probability in the rounded
solution. The authors consider this a very interesting open problem.
Another interesting possibility would be the generalisation of the forcing constraints of
eMWMBG or the general k-ePLP model to priorities. In algorithm PEPS, all the relaxed
terms are pushed with the same weight to become tight. Using different weights for each
constraint could lead the constraints to become more or less tight depending on their weightpriority. For matching this could lead to a kind of a priority matching algorithm.
The solutions to the relaxed decision procedures can be of practical interest even when
the problem is considered unfeasible in the algorithm. The algorithm returns a solution vector
for k-ePLP that contains important information about which constraints could be made tight
or almost tight. It is interesting to consider if this feature can be exploited in heuristics for
practical problems.
The PEPS methodology presented in this work is a flexible technique that can be proven
useful for other parallel algorithms. The relaxation of “hard” problem constraints and the
reduction of the complex optimization problem to its search or decision version, are two
powerful tools for the design of parallel approximation algorithms. Even though PEPS was
presented in this work on PLP in packing form it achieves equivalent results for covering
form PLP. As an illustrative application, an extension of Minimum Set Cover can be defined
where up to polylog sets can be restricted to be covered exactly once. PEPS achieves an
NCAS for the ε-relaxed version of this problem. Due to luck of space we did not include this
in the present paper.
The final result that connects FNCAS for PLP with an NC algorithm for MMBG relates
two open problems in the field of parallel complexity. PLP can easily model a set of problems
and this could be used to relarte more open problems in this field. In [TX98] it is shown that
the reduction used to prove P-Completeness of PLP was not appropriate to prove PCompleteness for the FNCAS for PLP. The counterexample that is used in [TX98] is based on
a boolean circuit of linear length. We conjecture that specific subclasses of CVP can be
reduced to an FNCAS for PLP. Such a subclass might be the boolean circuits of polylog depth
and polynomial size. The running time for CVP reduced to an FNCAS for PLP should be
polynomial on the depth and logarithmic on the size of the boolean circuit.
8
Acknowledgements
We would like to thank Dimitris Fotakis for useful discussions and his comments on an
earlier version of the paper. We are also grateful to Luca Trevisan and Fatos Xhafa whose
work [Tr96], [Tr98] and [TX98] inspired part of our research.
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