Network Design and
Bidimensionality
Mohammad T. Hajiaghayi
University of Maryland
Outline
Buy-at-bulk
Network Design
 Prize-collecting
Network Design
 Bidimensionality Theory
Steiner Trees
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Defined by Gauss in 1836
Given a graph and a subset of
nodes, find a subgraph that
connects these nodes
(e.g., clients and a server)
Objective: Minimize the total
connection cost
(e.g., cable installation cost)
NP-hard [Garey and Johnson’79]
Different from Minimum Spanning Trees:
Intermediate nodes
Approximating the Optimal Steiner Tree
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Approximation:
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Importance of Approximation Algorithms?
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Measured by its approximation factor, the ratio between the
approximate cost and the cost of an optimal solution
Approximation factors are worst-case bounds; practical
performance is often much better
Can be combined with other heuristics, like local search
Give better understanding to design heuristics
Provide provable lower bounds on optimum
The best approximation factor for Steiner trees is 1.38
[BGRS’10]
Steiner Forests
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More generally, connecting a set of pairs (e.g. multiple
servers for multiple VPNs)
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5
Objective: Minimize total
connection cost
Solution is a forest,
not necessarily a tree
The best approximation factor
is 2 by a greedy algorithm
[AKR’91, GW’95]
3
21
9
7
14
8
16
12
Let’s see a generalization with profound
practical applications in telecommunication
(e.g., at AT&T, Bell-labs)
2
21
27
5
Buy-in-Bulk Generalization
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Buying bandwidth to meet demands between a set of
pairs of nodes
Cost of buying bandwidth satisfies economies of scale
Different cable types like T1,T2,T3, OS12, OS48, etc.
Capacity on a link can be purchased
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at discrete units: u1   ur
with associated costs: c1   cr
c
c
where: 1   r (economies of scale)
u1
ur
So, if you buy in bulk, you save
Generalization (cont’d)
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A non-decreasing monotone concave (or generally
sub-additive) function fe: R+
R+ for an edge e where
fe(b) is the minimum cost of cable installation with
bandwidth b for edge e
fe(b)
cost
bandwidth
Multi-Commodity
Buy-at-Bulk (MC-BB) : Given a
set of bandwidth demand pairs,
install sufficient capacities at
minimum total cost
Cost-Distance
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Multi-commodity buy-at-bulk is equivalent
to the cost-distance problem
(up to a factor 1+ ε):
On each edge
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cost function (installation cost) c: E
R+
length function (per-use routing cost) l : E
R+
Also a set of pairs (si , ti) of nodes with a
traffic demand di between them
Goal: minimize total cost of installation plus routing
Cost-Distance (more formally)
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Feasible solution: a subset E ' of E such that all pairs
si , ti are connected in G [E ']
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Cost of the solution:
h
c( E ')   d i lE  ( si , ti )
i 1
where lE ' (si , ti) is the shortest l- path in G [E ']
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Goal: minimize total cost
Example
Contribution of this edge to total cost is
14+2*1=16.
c=14
Contribution of this
edge to total cost is
0+2*3=6
l=1
c=0
l=3
10
All demands di =1
10
Special Cases
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Single-source (SS-BB) case: all si (sources)
are equal
Uniform case: cost and length
Single-source
functions on edges are all
the same, i.e. , each edge e has
cost c + l demand-passing(e)
for constants c and l
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12
8
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Algorithms for Special Cases
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O(log n) approximation algorithms for special cases:
Single source:
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[Guha, Meyerson, and Munagala ’01]:
[Talwar ’02]
[Gupta, Kumar, and Roughgarden ’02]
[Meyerson, Munagala, and Plotkin ’00]
[Goel and Estrin ’03]
[Chekuri, Khanna, and Naor ’01]
…
Uniform multicommodity:
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[Awerbuch and Azar ’97]
[Bartal ’98]
[Gupta, Kumar, Pal, and Roughgarden ’03]
…
Almost logarithmic hardness in these cases [Andrews ’04].
But no algorithm with good (e.g. polylogarithmic) approximation factor for the
most general multi-commodity (non-uniform) buy-at-bulk case for over a decade
12
Our Main Result
[Chekuri, Hajiaghayi, Kortsarz, Salavatipour, FOCS’06, SICOMP’10]
Theorem: For h number of si , ti pairs, we obtain a (practical)
polynomial-time algorithm with approximation ratio O(log4 h).
For simplicity, will present the unit-demand case
(i.e. di=1 for all i’s) and present Õ(log4 n)
13
Overview of the Algorithm
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The algorithm iteratively finds a partial solution
connecting some of the residual pairs
The pairs are then removed from the set; repeat until
all pairs are connected (routed)
Density of a partial solution =
cost of the partial solution
# of new pairs routed
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Density is the average cost per new routed pair
The algorithm tries to find a low density partial
solution at each iteration
Overview of the Algorithm (cont’d)
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Will show the density of each partial solution in our
algorithm is at most Õ(log3 n)  (OPT / h') where
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OPT is the cost of optimum solution
h' is the number of unrouted pairs
A simple analysis (like for set cover) shows:
Total Cost
 Õ(log3 n)  OPT  (1/n2 + 1/(n2 - 1) +…+ 1)
 Õ(log4 n)  OPT
Structure of (near) Optimum
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How to compute a low-density partial solution?
Prove the existence of low-density one with a very
specific structure: junction tree
Junction tree: given a set P of pairs, tree T rooted at r is a
junction tree if
r
 It contains all pairs of P
 For every pair si , ti P
the path connecting
them in T goes through r
Why junction trees? knowing the pairs reduces the problem
to single-source buy-at-bulk (SS-BB) (with O(log n) approx.)
Summary of the Algorithm
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So two main ingredients in the proof
Theorem 2: There is always a partial solution that
is a junction tree with density Õ (log n)  (OPT / h')
Theorem 3: There is an O (log2 n) approximation
for finding lowest density junction tree
(this is low density SS-BB).
Corollary: We can find a partial solution with
density Õ (log3 n)  (OPT / h')
This implies an approximation Õ (log4 n) for MC-BB
Notations for Proof of Thm 2
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We provide a junction tree partial solution with
density Õ (log n)  (OPT / h')
Consider an optimum solution OPT
Let
 E* be the edge set that OPT installs,
 OPTc be its (installation) cost
 OPTl be the total length (per-use routing cost).
Thus OPT= OPTc + OPTl
Removing Cycles
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OPT may have cycles !
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By [Elkin, Emek, Spielman, and Teng ’05, Abraham, Bartal,
Neiman’08] on probabilistic embedding on spanning trees
and by losing a factor Õ (log n) on length, we can assume
E* is a forest T
(WLOG assume T is connected).
Junction Tree with Low Density
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From T we obtain a collection of rooted subtrees (in
the form of junction trees) T1,…,Ta such that
 any edge e of T is included in at most O(log n) of
subtrees
 for every pair there is exactly one index i such that
both vertices are in Ti and their path in Ti goes
through the root of Ti
The total cost of the junction trees is at most
O (log n)  OPTc + Õ (log n)  OPTl =Õ (log n)  OPT
Thus at least one of junction trees of T1,…,Ta has the
desired density of Õ (log n)  (OPT / h')
Decomposition into Junction Trees
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Given T, pick a centroid r1 (i.e., largest remaining
component has at most (2/3) |V(T)| vertices)
Add tree T rooted at r1 to the collection and the pairs
whose paths go through r1
Remove r1 from T and apply the procedure recursively
to each of the resulting component
Each pair is on exactly one subtree
in the collection
 Each edge is on O (log n) subtrees since
the depth of recursion is O (log n)
We are done with the first main theorem
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r2
21
r1
Details of Proof of Thm 3
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Theorem 3: There is an O (log2 n) approximation for
finding lowest density junction tree
Very similar to single-source except that we have to find
a lowest density solution
Goal: connect a subset of pairs to
r
the root r with lowest density
(= cost of solution / # of pairs in sol)
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Formulate the problem as an
Integer Programming (IP) and then consider
the Linear Programming (LP) relaxation
First Low Density Single-Sink
r
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Let T be set of terminals to be connected to r
yi is one if we connect terminal i to r
x(e) is one if the edge is in our solution
Let Pi be set of paths from terminal i to r
fp is the flow on path p
Above IP denotes the lowest density (lowest average cost)
way of connecting a set of terminals from T to r
Finding Low Density Junction Tree
r
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Solve the above LP and partition the terminals of T into log n classes
[1-1/2], [1/2-1/4], [1/4-1/8], … with almost equal y variable
Find a class S of terminals among log n classes with max sum of y
variables and scale up (lose a factor O (log n))
Use O (log n) approx of [MMP’00,CKN’01] for SS-BB on S
Some Recent Extensions
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O(log3n) approx for non-uniform buy at bulk when demands are polynomial [Kortsarz
and Nutov’ 07]
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O(log4n) approx can be extended to the node-weighted case but requires some new
ideas and some extra work [Chekuri, Hajiaghayi, Kortsarz, Salavatipour ‘07]
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O(log4n) approx when want to have two disjoint paths between each demand pair
[Chekuri, Antonakapoulos, Shepherd and Zhang’ 11]
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O(n1/2) approx for the multicommodity case in directed graphs
[Chekuri, Even, Gupta, and Segev’ 08]
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Our results can be extended to stochastic Steiner tree with non-uniform inflation (by
losing an extra factor O(log n)) [Gupta, Hajiaghayi, and Kumar ’07]
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Same technique has been used in the Dial-a-Ride problem
[Gupta, Hajiaghayi, Ravi, and Nagarajan ’07]
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Oblivious network design with ratio O(log3 n) for uniform buy-at-bulk, i.e., costs of all
edges are the same sub-additive function f [Gupta, Hajiaghayi, and Raecke ’07]
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Currently thinking of Capacitated Network Design
Prize-collecting Network Design
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Prize-collecting problems: classic optimization problems with
various demands to be ``served'' by some lowest-cost
structure
However, if some demands are too expensive to serve, then
refuse and instead pay a penalty
Several applications both in
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Theory: Game theory, Lagrangian relaxation
Practice: Real-world AT&T application saving millions of dollar in
design of fiber networks
Studies for several problems, e.g., [B’89, GW’92, HJ’06,
CRR’99,KNN’10,BHM’11,BH’10,HN’10,ABHK’11]
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Prize-collecting Steiner Trees (PCST)
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Given: graph G=(V, E), edge costs ce ≥ 0, root r, penalties
pv ≥ 0 on vertices
Goal: choose subtree T so as to cost of edges in T + penalty of
nodes not connected to r, i.e., ∑e in T ce + ∑v not connected to r pv , is
minimized
r
Tree T
AT&T Application: Design fiber build
connecting new customers to existing net.
Graph: street network
Root: existing fiber (supernode)
Edge cost: digging trench and laying fiber
Prize: monthly income for each new
customer
Our Improvement
[Archer, Bateni, Hajiaghayi, Karloff, FOCS’09, SICOMP’11]
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Balas’89: introduce PCST
Bienstock et al.’93: give 3-approx. LP-rounding
Goemans-Williamson’92 : 2-approx primal-dual.
Several other heuristics since then [CRR’99,LR’00]
Improving on factor 2 was a famous open problem for 17
years
We obtain 1.967-approx for PCST problems via a
Prize-Collecting Clustering technique
Why is it important?
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Breaking the barrier and open the path for others
A little improvement (e.g. 2%) can save a lot of money in practice
Technique is new and exciting
Prize-Collecting Clustering
[Bateni, Hajiaghayi, Marx, STOC’10, J. ACM]
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New clustering paradigm based on
prize-collecting frameworks
Cluster vertices of a graph
each have a budget
A cluster: a tree
connecting its vertices
Connecting cost of a cluster payable by budgets of its
vertices
Cost of connecting different clusters not payable by
their budgets
PC-Clustering Applications:
1. PC Steiner tree: 1.967-approx[Archer, Bateni, Hajiaghayi,
Karloff ’09]
2. PCTSP (and Tour): 1.980- approx[Archer, Bateni,
Hajiaghayi, Karloff ’09]
3. Planar Steiner forest: PTAS (1+ε)[Bateni, Hajiaghayi,
Marx’10]
4. Planar submodular prize-collecting Steiner forest:
Reduction to bounded-treewidth graphs[Bateni, Chekuri,
Ene, Hajiaghayi, Korula, Marx’11]
5. Planar multiway cut: PTAS (1+ε) [Bateni, Hajiaghayi, Klein,
Mathieu’12] improving over factor 1.34 for general graphs
Bidimensionality Theory
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Main (theoretical) approaches to solve NP-hard network design
problems:
 Special instances: Planar graphs, bounded genus graphs (fiber
networks in ground), etc.
 Approximation algorithms (PTAS):
Within a factor C of the optimal solution
(PTAS if C= 1+ ε for arbitrary constant ε)
 Fixed-parameter algorithms:
Parameterize problem by parameter P
(typically, the cost of the optimal solution)
and aim for f(P) nO(1) (or even f(P) + nO(1))
We consider all above in Bidimensionality and aim for general
algorithmic frameworks
Overview
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For any network design problem in a large class
(“bidimensional”)
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Vertex cover, dominating set, connected dominating set, rr
dominating set, feedback vertex set, TSP, k-cut, Steiner tree, Steiner
forest, multiway cut,…
In broad classes of networks generalizing planar networks
(most “minor-closed” graph families)
We obtain (in a series of more than 25 papers):
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Strong combinatorial properties
Fixed-parameter algorithms
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Often subexponential: 2O(√k) nO(1) where k=|OPT|
Approximation algorithms
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Often PTASs (1+ ε approx): f(1/ε) nO(1)
r
Summary of Results
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A general algorithmic framework (with Rajesh)
Introducing the concept of graph contraction instead of
graph minor
Simplifying network decompositions: decompose networks
into algorithmically simple instances instead of necessarily
small-size networks
Improving deep graph-minor theory of Robertson-Seymour
and make it algorithmic
Three workshops so far on the theory Berlin (2007),
Dagstuhl (2009), Dagstuhl (2013)