Approximation algorithms for network design: A survey

Abstract

In a typical instance of a network design problem, we are given a directed or undirected graph G = (V, E), non-negative edge-costs c e for all e ∈ E, and our goal is to find a minimum-cost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimum-cost set of edges that induces a connected graph (this is the minimum-cost spanning tree problem), or we might want to find a minimum-cost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimum-cost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples.Many practically relevant instances of network design problems are NP-hard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomial-time), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an α-approximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the following we will also sometimes refer to α as the performance guarantee of the respective approximation algorithm.The last 30 years have seen a tremendous amount of research on approximation algorithms for network design problems. And over this period, several technical themes have emerged, and have been explored and exploited to give algorithms and analyze their performance. Our aim in this survey is to provide an overview over these techniques. Each of the following sections focuses on one technique and has two main parts: first, we present an introductory application to the well-known classical minimum-spanning tree problem. The second part of each section demonstrates more sophisticated recent example applications of the respective technique. Throughout we assume that the reader is familiar with fundamental concepts of graph theory, combinatorial optimization, and approximation algorithms. While we may recap certain key definitions, we rely on the reader to be familiar with others. We refer to the excellent text books [45,160,163] for background reading.The minimum spanning tree problem has been studied for at least a century, and it is clearly one of the most prominent network design problems. The input to an instance of this problem consists of an undirected graph G = (V, E) each of whose edges e ∈ E is endowed by an arbitrary cost c e , and the goal is to compute a spanning tree of smallest cost. The earliest known algorithm for this problem was developed by Boruvka [21], and since then a vast number of techniques have...

The Interdependent Network Design Problem for Optimal Infrastructure System Restoration

“…Gupta andKönemann (2011) presented a review of diverse approximation algorithms that take advantage of the structure and properties of the NDP, while showing tailor made heuristic approaches. On the other hand, Chen et al (2011) provided a review of approaches and applications of the NDP under uncertainty, with a main focus on studying the reliability of networks related to transportation.…”

FireFly

“…From this view, the SM-PCFT problem falls in the class of network design problems [25], but is different from prior work due to the novel DBW objective. For instance, even the special case of k = 1, the SM-PCFT problem reduces to constructing an m-regular graph over n nodes with maximum (static) bisection bandwidth.…”

Primal-dual approximation algorithms for Node-Weighted Steiner Forest on planar graphs

“…The problem Node-Weighted Steiner Forest is to find a minimum weight set X ⊆ V of vertices such that for all 1 i k, s i and t i are in the same component of G [ X]. The usual undirected cut-formulation of the problem as integer linear program is the following [19,14,31,23]. Let T be the family of cuts that separate a terminal pair:…”

“…To apply the same algorithm to NodeWeighted Steiner Forest, Demaine et al [14] introduced a slight modification in the initialization, that we describe here. For more information on primal-dual algorithms, we refer to the surveys [19,23] and the exposition in [31]. We remark that the primal-dual algorithm for Steiner problems can also be reformulated as a local ratio algorithm [4].…”