Discrete optimization problems are everywhere, from traditional operations research planning problems, such as scheduling, facility location, and network design; to computer science problems in databases; to advertising issues in viral marketing. Yet most such problems are NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal solutions to such problems. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first part of the book is devoted to a single algorithmic technique, which is then applied to several different problems. The second part revisits the techniques but offers more sophisticated treatments of them. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithms courses, the book will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

https://www.designofapproxalgs.com/book.pdf

The Design of Approximation Algorithms

The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e

/pdf/an-improved-lp-based-approximation-for-steiner-tree-4measlqfh9.pdf

An improved LP-based approximation for steiner tree

The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999].In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + e

/pdf/steiner-tree-approximation-via-iterative-randomized-rounding-munhd0nqsg.pdf

Steiner Tree Approximation via Iterative Randomized Rounding

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs.

/pdf/designing-network-protocols-for-good-equilibria-2jv95cwie7.pdf

Designing Network Protocols for Good Equilibria

In this paper, we study online algorithms when the input is not chosen adversarially, but consists of draws from some given probability distribution. While this model has been studied for online problems like paging and k-server, it is not known how to beat the Φ(log n) bound for online Steiner tree if at each time instant, the demand vertex is a uniformly random vertex from the graph. For the online Steiner tree problem, we show that if each demand vertex is an independent draw from some probability distribution π: V → [0, 1], a variant of the natural greedy algorithm achieves Eω[A(ω)]/Eω[OPT (ω)] = O(1); moreover, this result can be extended to some other subadditive problems. Both assumptions that the input sequence consists of independent draws from π, and that π is known to the algorithm are both essential; we show (almost) logarithmic lower bounds if either assumption is violated. Moreover, we give preliminary results on extending the Steiner tree results above to the related "expected ratio" measure Eω[ω(ω)/OPT (ω)]. Finally, we use these ideas to give an average-case analysis of the Universal TSP problem.

/pdf/stochastic-analyses-for-online-combinatorial-optimization-5e2v4vxu0d.pdf

Stochastic analyses for online combinatorial optimization problems

We present constant-factor approximation algorithms for several widely-studied NP-hard optimization problems in network design, including the multicommodity rent-or-buy, virtual private network design, and single-sink buy-at-bulk problems. Our algorithms are simple and their approximation ratios improve over those previously known, in some cases by orders of magnitude.We develop a general analysis framework to bound the approximation ratios of our algorithms. This framework is based on a novel connection between random sampling and game-theoretic cost sharing.

/pdf/approximation-via-cost-sharing-simpler-and-better-4snj6j8j0w.pdf

Martin P´al

Papers

Approximation via cost sharing: Simpler and better approximation algorithms for network design