Greedy Algorithms for Online Survivable Network Design

In an instance of the network design problem, we are given a graph G=(V,E), an edge-cost function c:E -> R^{>= 0}, and a connectivity criterion. The goal is to find a minimum-cost subgraph H of G that meets the connectivity requirements. An important family of this class is the survivable network design problem (SNDP): given non-negative integers r_{u v} for each pair u,v in V, the solution subgraph H should contain r_{u v} edge-disjoint paths for each pair u and v. While this problem is known to admit good approximation algorithms in the offline case, the problem is much harder in the online setting. Gupta, Krishnaswamy, and Ravi [Gupta et al., 2012] (STOC\u2709) are the first to consider the online survivable network design problem. They demonstrate an algorithm with competitive ratio of O(k log^3 n), where k=max_{u,v} r_{u v}. Note that the competitive ratio of the algorithm by Gupta et al. grows linearly in k. Since then, an important open problem in the online community [Naor et al., 2011; Gupta et al., 2012] is whether the linear dependence on k can be reduced to a logarithmic dependency. Consider an online greedy algorithm that connects every demand by adding a minimum cost set of edges to H. Surprisingly, we show that this greedy algorithm significantly improves the competitive ratio when a congestion of 2 is allowed on the edges or when the model is stochastic. While our algorithm is fairly simple, our analysis requires a deep understanding of k-connected graphs. In particular, we prove that the greedy algorithm is O(log^2 n log k)-competitive if one satisfies every demand between u and v by r_{uv}/2 edge-disjoint paths. The spirit of our result is similar to the work of Chuzhoy and Li [Chuzhoy and Li, 2012] (FOCS\u2712), in which the authors give a polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2. Moreover, we study the greedy algorithm in the online stochastic setting. We consider the i.i.d. model, where each online demand is drawn from a single probability distribution, the unknown i.i.d. model, where every demand is drawn from a single but unknown probability distribution, and the prophet model in which online demands are drawn from (possibly) different probability distributions. Through a different analysis, we prove that a similar greedy algorithm is constant competitive for the i.i.d. and the prophet models. Also, the greedy algorithm is O(log n)-competitive for the unknown i.i.d. model, which is almost tight due to the lower bound of [Garg et al., 2008] for single connectivity

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity

Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

Online Network Design under Uncertainty

Today, computer and information networks play a significant role in the success of businesses, both large and small. Networks provide access to various services and resources to end users and devices. There has been extensive research on de- signing networks according to numerous criteria such as cost-efficiency, availability, adaptivity, survivability, among others. In this dissertation, we revisit some of the most fundamental network design problems in the presence of uncertainty. In most realistic models, we are forced to make decisions in the presence of an incomplete input, which is the source of uncertainty for an optimization algorithm. There are different types of uncertainty. For example, in stochastic settings, we may have some random variables derived from some known/unknown distributions. In online settings, the complete input is not known in a-priori and pieces of the input become available sequentially; leaving the algorithm to make decisions only with partial data. In this dissertation, we consider network design and network optimization problems with uncertainty. In particular, we study online bounded-degree Steiner network design, online survivable network design, and stochastic k-server. We analyze their complexity and design competitive algorithms for them

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

Online Decision Making via Prophet Setting

In the study of online problems, it is often assumed that there exists an adversary who acts against the algorithm and generates the most challenging input for it. This worst-case assumption in addition to the complete uncertainty about future events in the traditional online setting sometimes leads to worst-case scenarios with super-constant approximation impossibilities. In this dissertation, we go beyond this worst-case analysis of problems by taking advantage of stochastic modeling. Inspired by the prophet inequality problem, we introduce the prophet setting for online problems in which the probability distributions of the future inputs are available. This modeling not only considers the availability of statistical data in the design of mechanisms but also results in significantly more efficient algorithms. To illustrate the improvements achieved by this setting, we study online problems within the contexts of auctions and networks. We begin our study with analyzing a fundamental online problem in optimal stopping theory, namely prophet inequality, in the special cases of iid and large markets, and general cases of matroids and combinatorial auctions and discuss its applications in mechanism design. The stochastic model introduced by this problem has received a lot of attention recently in modeling other real-life scenarios, such as online advertisement, because of the growing ability to fit distributions for user demands. We apply this model to network design problems with a wide range of applications from social networks to power grids and communication networks. In this dissertation, we give efficient algorithms for fundamental network design problems in the prophet setting and present a general framework that demonstrates how to develop algorithms for other problems in this setting

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions