Showing papers on "Set cover problem published in 1998"

Greedy strikes back: improved facility location algorithms

Abstract: A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as theuncapacitated facility locationproblem. Application to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser, and Wolsey. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos, and Aardal. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos, and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is max SNP-hard. However, the inapproximability constants derived from the max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio, assumingNP?DTIMEnO(loglogn)].

Improved Methods for Approximating Node Weighted Steiner Trees and Connected Dominating Sets

Abstract: In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on spider decompositions was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2ln κ, where κ is the number of terminals. However, the best known lower bound on the approximation ratio is (1 - o(1)) ln κ 1 assuming that NP DTIME[n O(log log n) ], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of In κ. For the weighted case we develop a new decomposition theorem, and generalize the notion of spiders to branch-spiders, that are used to design a new algorithm with a worst case approximation factor of 1.5 ln κ. We then generalize the method to yield an approximation factor of (1.35+∈) ln κ, for any constant e > 0. These algorithms, although polynomial, are not very practical due to their high running time; since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln κ. The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0-1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n due to Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35 + ∈) ln n for any fixed ∈ > 0.

Approximation algorithms for set cover and related problems

Abstract: In this thesis, we analyze several known and newly designed algorithms for approximating optimal solutions to NP-hard optimization problems. We give a new analysis of the greedy algorithm for approximating the S scET C scOVER and P scARTIAL S scET scCOVER problems obtaining significantly improved performance bounds. We also give a first approximation algorithm with a non-trivial performance bound for the E scRRAND S scCHEDULING and T scREE C scOVER problems, known also as the G scENERALIZED T scRAVELING S scALESMAN and G scROUP S scTEINER T scREE problems. The main results of this thesis first appeared in my papers (87), (89), (91), and (90); and in my technical reports (86) and (88).

Lectures on proof verification and approximation algorithms

Abstract: to the theory of complexity and approximation algorithms.- to randomized algorithms.- Derandomization.- Proof checking and non-approximability.- Proving the PCP-Theorem.- Parallel repetition of MIP(2,1) systems.- Bounds for approximating MaxLinEq3-2 and MaxEkSat.- Deriving non-approximability results by reductions.- Optimal non-approximability of MaxClique.- The hardness of approximating set cover.- Semidefinite programming and its applications to approximation algorithms.- Dense instances of hard optimization problems.- Polynomial time approximation schemes for geometric optimization problems in euclidean metric spaces.

Max Horn SAT and the minimum cut problem in directed hypergraphs

Abstract: In this paper we consider the Maximum Horn Satisfiability problem, which is reduced to the problem of finding a minimum cardinality cut on a directed hypergraph. For the latter problem, we propose different IP formulations, related to three different definitions of hyperpath weight. We investigate the properties of their linear relaxations, showing that they define a hierarchy. The weakest relaxation is shown to be equivalent to the relaxation of a well known IP formulation of Max Horn SAT, and to a max-flow problem on hypergraphs. The tightest relaxation, which is a disjunctive programming problem, is shown to have integer optimum. The intermediate relaxation consists in a set covering problem with a possible exponential number of constraints. This latter relaxation provides an approximation of the convex hull of the integer solutions which, as proven by the experimental results given, is much tighter than the one known in the literature. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An Iterated Heuristic Algorithm for the Set Covering Problem

Abstract: The set covering problem is a well-known NP-hard combinatorial optimization problem with wide practical applications. This paper introduces a novel heuristic for the uni-cost set covering problem. An iterated approximation algorithm (ITEG) based on this heuristic is developed: in the rst iteration a cover is constructed, and in the next iterations a new cover is built by starting with part of the best solution so far obtained. The nal output of the algorithm is the best solution obtained in all the iterations. ITEG is empirically evaluated on a set of randomly generated problem instances, on instances originated from the Steiner triple systems, and on instances derived from two challenging combinatorial questions by Erdd os. The performance of ITEG on these benchmark problems is very satisfactory, both in terms of solution quality (i.e., small covers) as well as in terms of running time.

Automatic test generation of linear analog circuits under parameter variations

Abstract: This paper presents a simulation-based approach to automatic test-frequency generation of linear analog circuits considering parameter variations. It consists of two steps. First, a candidate set of frequencies-each detects robustly some faults-is generated by a concurrent and lazy fault simulation method. Next, the minimum number of test frequencies to detect all the possible faults is selected by solving the set covering problem. With an interval-mathematic algorithm to computer circuit responses under parameter variations and a decision-diagram-based algorithm for exact set covering, the proposed approach is fast and capable of finding an optimal set of test frequencies. The approach has been implemented, and some experimental results are described.

Set Covering Algorithms in Edit Generation

Abstract: Results are presented from a comparison study of several set covering algorithms (routines) used in the implicit edit generation algorithms of Gar nkel, Kunnathur, and Liepins [1986] and Winkler [1997]. The edit generation algorithms are based on the Fellegi and Holt model [1976] of editing. Since the set covering routine is called many times in edit generation, an e cient routine will signi cantly reduce the computing time of the generation process. Unlike most of the applications of the set covering problem (SCP), in which an optimal cover is desirable, the edit generation is interested in nding all the prime covers to a SCP.

One for the Price of Two: A Unified Approach for Approximating Covering Problems

Abstract: We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest and related problems.