Network Design and Bidimensionality

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Network Design and Bidimensionality

• Mohammad T. Hajiaghayi
• University of Maryland

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Outline

• Buy-at-bulk Network Design
• Prize-collecting Network Design
• Bidimensionality Theory

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Steiner Trees

• 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 Johnson79
• Different from Minimum Spanning Trees
  Intermediate nodes

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Approximating the Optimal Steiner Tree

• Approximation
• Measured by its approximation factor, the ratio
  between the approximate cost and the cost of an
  optimal solution
• Importance of Approximation Algorithms?
• 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 BGRS10

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Steiner Forests

• More generally, connecting a set of pairs (e.g.
  multiple servers for multiple VPNs)
• Objective Minimize total connection cost
• Solution is a forest, not necessarily a tree
• The best approximation factor is 2 by a greedy
  algorithmAKR91, GW95
• Lets see a generalization with profound
• practical applications in telecommunication
  (e.g., at ATT, Bell-labs)

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Buy-in-Bulk Generalization

• 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
• at discrete units
• with associated costs
• where (economies of scale)
• So, if you buy in bulk, you save

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Generalization (contd)

• 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)
Multi-Commodity Buy-at-Bulk (MC-BB) Given a
set of bandwidth demand pairs, install sufficient
capacities at minimum total cost
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Cost-Distance

• Multi-commodity buy-at-bulk is equivalent
• to the cost-distance problem
• (up to a factor 1 e)
• On each edge
• 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

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Cost-Distance (more formally)

• Feasible solution a subset E ' of E such
  that all pairs
• si , ti are connected in G E '
• Cost of the solution
• where lE ' (si , ti) is the shortest l-
  path in G E '
• Goal minimize total cost

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Example
Contribution of this edge to total cost is
142116.
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c14
Contribution of this edge to total cost is 0236
l1
c0
l3
All demands di 1
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Special Cases

• Single-source (SS-BB) case all si (sources)
• are equal
• Uniform case cost and length
• functions on edges are all
• the same, i.e. , each edge e has
• cost c l? demand-passing(e)
• for constants c and l

Single-source
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Algorithms for Special Cases

• O(log n) approximation algorithms for special
  cases
• Single source
• 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
• 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

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Our Main Result Chekuri, Hajiaghayi,
Kortsarz, Salavatipour, FOCS06, SICOMP10

• 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. di1 for all is) and present Õ(log4 n)

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Overview of the Algorithm

• 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
• Density is the average cost per new routed pair
• The algorithm tries to find a low density partial
  solution at each iteration

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Overview of the Algorithm (contd)

• Will show the density of each partial solution in
  our algorithm is at most Õ(log3 n) ? (OPT / h')
  where
• 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

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Structure of (near) Optimum

• 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
• 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.)

r
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Summary of the Algorithm

• 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

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Notations for Proof of Thm 2

• 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

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Removing Cycles

• OPT may have cycles !
• By Elkin, Emek, Spielman, and Teng 05, Abraham,
  Bartal, Neiman08 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).

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Junction Tree with Low Density

• 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')

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Decomposition into Junction Trees

• 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

r1
r2
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Details of Proof of Thm 3

• 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
• the root r with lowest density
• ( cost of solution / of pairs in sol)
• Formulate the problem as an
• Integer Programming (IP) and then consider
• the Linear Programming (LP) relaxation

r
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First Low Density Single-Sink
r

• 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

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Finding Low Density Junction Tree
r

• 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 MMP00,CKN01 for
  SS-BB on S

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Some Recent Extensions

• O(log3n) approx for non-uniform buy at bulk when
  demands are polynomial Kortsarz and Nutov 07
• 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
• O(log4n) approx when want to have two disjoint
  paths between each demand pair Chekuri,
  Antonakapoulos, Shepherd and Zhang 11
• O(n1/2) approx for the multicommodity case in
  directed graphs
• Chekuri, Even, Gupta, and Segev 08
• 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
• Same technique has been used in the Dial-a-Ride
  problem
• Gupta, Hajiaghayi, Ravi, and Nagarajan 07
• 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
• Currently thinking of Capacitated Network Design

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Prize-collecting Network Design

• 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
• Theory Game theory, Lagrangian relaxation
• Practice Real-world ATT application saving
  millions of dollar in design of fiber networks
• Studies for several problems, e.g., B89, GW92,
  HJ06, CRR99,KNN10,BHM11,BH10,HN10,ABHK11

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Prize-collecting Steiner Trees (PCST)

• 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

ATT 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
r
Tree T
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Our Improvement Archer, Bateni, Hajiaghayi,
Karloff, FOCS09, SICOMP11

• Balas89 introduce PCST
• Bienstock et al.93 give 3-approx. LP-rounding
• Goemans-Williamson92 2-approx primal-dual.
• Several other heuristics since then
  CRR99,LR00
• 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?
• 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

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Prize-Collecting Clustering Bateni, Hajiaghayi,
Marx, STOC10, J. ACM

• 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

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PC-Clustering Applications

1. PC Steiner tree 1.967-approxArcher, Bateni,
   Hajiaghayi, Karloff 09
2. PCTSP (and Tour) 1.980- approxArcher, Bateni,
   Hajiaghayi, Karloff 09
3. Planar Steiner forest PTAS (1e)Bateni,
   Hajiaghayi, Marx10
4. Planar submodular prize-collecting Steiner
   forest Reduction to bounded-treewidth
   graphsBateni, Chekuri, Ene, Hajiaghayi, Korula,
   Marx11
5. Planar multiway cut PTAS (1e) Bateni,
   Hajiaghayi, Klein, Mathieu12 improving over
   factor 1.34 for general graphs

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Bidimensionality Theory

• 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 e for arbitrary constant e)
• Fixed-parameter algorithmsParameterize 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

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Overview

• For any network design problem in a large class
  (bidimensional)
• Vertex cover, dominating set, connected
  dominating set, r-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)
• Strong combinatorial properties
• Fixed-parameter algorithms
• Often subexponential 2O(vk) nO(1) where kOPT
• Approximation algorithms
• Often PTASs (1 e approx) f(1/e) nO(1)

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Summary of Results

• 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)

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