Network Design and Bidimensionality Mohammad T Hajiaghayi University

Network Design and Bidimensionality Mohammad T. Hajiaghayi University of Maryland

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

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 Johnson’ 79] Different from Minimum Spanning Trees: Intermediate nodes

Approximating the Optimal Steiner Tree Approximation: Importance of Approximation Algorithms? 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 More generally, connecting a set of pairs (e. g. multiple servers for multiple VPNs) 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 Buying bandwidth to meet demands between a set of pairs of nodes Cost of buying bandwidth satisfies economies of scale Different cable types like T 1, T 2, T 3, OS 12, OS 48, 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

Generalization (cont’d) 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 Multi-commodity buy-at-bulk is equivalent to the cost-distance problem (up to a factor 1+ ε): 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

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 l. E ' (si , ti) is the shortest l- path in G [E '] 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 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 8 21 12 11 5

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 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(log 4 h). For simplicity, will present the unit-demand case (i. e. di=1 for all i’s) and present Õ(log 4 n) 13

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

Overview of the Algorithm (cont’d) Will show the density of each partial solution in our algorithm is at most Õ(log 3 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(log 3 n) OPT (1/n 2 + 1/(n 2 - 1) +…+ 1) Õ(log 4 n) OPT

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 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 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 (log 2 n) approximation for finding lowest density junction tree (this is low density SS-BB). Corollary: We can find a partial solution with density Õ (log 3 n) (OPT / h') This implies an approximation Õ (log 4 n) for MC-BB

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

Removing Cycles OPT may have cycles ! 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 From T we obtain a collection of rooted subtrees (in the form of junction trees) T 1, …, 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 T 1, …, Ta has the desired density of Õ (log n) (OPT / h')

Decomposition into Junction Trees Given T, pick a centroid r 1 (i. e. , largest remaining component has at most (2/3) |V(T)| vertices) Add tree T rooted at r 1 to the collection and the pairs whose paths go through r 1 Remove r 1 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 r 2 21 r 1

Details of Proof of Thm 3 Theorem 3: There is an O (log 2 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) Formulate the problem as an Integer Programming (IP) and then consider the Linear Programming (LP) relaxation

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

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 [MMP’ 00, CKN’ 01] for SS-BB on S

Some Recent Extensions O(log 3 n) approx for non-uniform buy at bulk when demands are polynomial [Kortsarz and Nutov’ 07] O(log 4 n) approx can be extended to the node-weighted case but requires some new ideas and some extra work [Chekuri, Hajiaghayi, Kortsarz, Salavatipour ‘ 07] O(log 4 n) approx when want to have two disjoint paths between each demand pair [Chekuri, Antonakapoulos, Shepherd and Zhang’ 11] O(n 1/2) 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(log 3 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

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 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] 26

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 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] 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? 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] 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 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) n. O(1) (or even f(P) + n. O(1)) We consider all above in Bidimensionality and aim for general algorithmic frameworks

Overview For any network design problem in a large class (“bidimensional”) Vertex cover, dominating set, connected dominating set, r- 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 r Often subexponential: 2 O(√k) n. O(1) where k=|OPT| Approximation algorithms Often PTASs (1+ ε approx): f(1/ε) n. O(1)

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 theory Berlin (2007), Dagstuhl (2009), Dagstuhl (2013)