Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009)

This issue of SICOMP contains nine specially selected papers from the Forty-first Annual ACM Symposium on the Theory of Computing, otherwise known as STOC 2009, held May 31 to June 2 in Bethesda, Maryland. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors, and subjected to the standard thorough reviewing process of SICOMP. The program committee consisted of Susanne Albers, Andris Ambainis, Nikhil Bansal, Paul Beame, Andrej Bogdanov, Ran Canetti, David Eppstein, Dmitry Gavinsky, Shafi Goldwasser, Nicole Immorlica, Anna Karlin, Jonathan Katz, Jonathan Kelner, Subhash Khot, Ravi Kumar, Leslie Ann Goldberg, Michael Mitzenmacher (Chair), Kamesh Munagala, Rasmus Pagh, Anup Rao, Rocco Servedio, Mikkel Thorup, Chris Umans, and Lisa Zhang. They accepted 77 papers out of 321 submissions

Online Network Design Algorithms via Hierarchical Decompositions

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.Comment: Accepted to SODA 201

Optimization problems in network connectivity

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 115-120).Besides being one of the principal driving forces behind research in algorithmic theory for more than five decades, network optimization has assumed increased significance in recent times with the advent and widespread use of a variety of large-scale real-life networks. The primary goal of such networks is to connect vertices (representing a variety of real-life entities) in a robust and inexpensive manner, and to store and retrieve such connectivity information efficiently. In this thesis, we present efficient algorithms aimed at achieving these broad goals. The main results presented in this thesis are as follows. -- Cactus Construction. We give a near-linear time Monte Carlo algorithm for constructing a cactus representation of all the minimum cuts in an undirected graph. -- Cut Sparsification. A cut sparsifier of an undirected graph is a sparse graph on the same set of vertices that preserves its cut values up to small errors. We give new combinatorial and algorithmic results for constructing cut sparsifiers. -- Online Steiner Tree. Given an undirected graph as input, the goal of the Steiner tree problem is to select its minimum cost subgraph that connects a designated subset of vertices. We give the first online algorithm for the Steiner tree problem that has a poly-logarithmic competitive ratio when the input graph has both node and edge costs. -- Network Activation Problems. In the design of real-life wireless networks, a typical objective is to select one among a possible set of parameter values at each node such that the set of activated links satisfy some desired connectivity properties. We formalize this as the network activation model, and give approximation algorithms for various fundamental network design problems in this model.by Debmalya Panigrahi.Ph.D

Solving two-stage stochastic network design problems to optimality

The Steiner tree problem (STP) is a central and well-studied graph-theoretical combinatorial optimization problem which plays an important role in various applications. It can be stated as follows: Given a weighted graph and a set of terminal vertices, find a subset of edges which connects the terminals at minimum cost. However, in real-world applications the input data might not be given with certainty or it might depend on future decisions. For the STP, for example, edge costs representing the costs of establishing links may be subject to inflations and price deviations. In this thesis we tackle data uncertainty by using the concept of stochastic programming and we study the two-stage stochastic version of the Steiner tree problem (SSTP). Thereby, a set of scenarios defines the possible outcomes of a random variable; each scenario is given by its realization probability and defines a set of terminals and edge costs. A feasible solution consists of a subset of edges in the first stage and edge subsets for all scenarios (second stage) such that each terminal set is connected. The objective is to find a solution that minimizes the expected cost. We consider two approaches for solving the SSTP to optimality: combinatorial algorithms, in particular fixed-parameter tractable (FPT) algorithms, and methods from mathematical programming. Regarding the combinatorial algorithms we develop a linear-time algorithm for trees, an FPT algorithm parameterized by the number of terminals, and we consider treewidth-bounded graphs where we give the first FPT algorithm parameterized by the combination of treewidth and number of scenarios. The second approach is based on deriving strong integer programming (IP) formulations for the SSTP. By using orientation properties we introduce new semi-directed cut- and flow-based IP formulations which are shown to be stronger than the undirected models from the literature. To solve these models to optimality we use a decomposition-based two-stage branch&cut algorithm, which is improved by a fast and efficient method for strengthening the optimality cuts. Moreover, we develop new and stronger integer optimality cuts. The computational performance is evaluated in a comprehensive computational study, which shows the superiority of the new formulations, the benefit of the decomposition, and the advantage of using the strengthened optimality cuts. The Steiner forest problem (SFP) is a related problem where sets of terminals need to be connected. On the one hand, the SFP is a generalization of the STP and on the other hand, we show that the SFP is a special case of the SSTP. Therefore, our results are transferable to the SFP and we present the first FPT algorithm for treewidth-bounded graphs and we model new and stronger (semi-)directed cut- and flow-based IP formulations for the SFP. In the second part of this thesis we consider the two-stage stochastic survivable network design problem, an extension of the SSTP where pairs of vertices may demand a higher connectivity. Similarly to the first part we introduce new and stronger semi-directed cut-based models, apply the same decomposition along with the cut strengthening technique, and argue the validity of the newly introduced integer optimality cuts. A computational study shows the benefit, robustness, and good performance of the decomposition and the cut strengthening method

Online and Stochastic Survivable Network Design

Consider the edge-connectivity survivable network design problem: given a graph G = (V, E) with edge-costs, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimum-cost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edge-connectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict cost-shares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rent-or-buy version and (b) the (twostage) stochastic version of the edge-connected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edge-costs are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)-strict cost shares, which gives constant-factor rent-or-buy and stochastic algorithms for these instances