Showing papers on "Set cover problem published in 2001"

An efficient distributed algorithm for constructing small dominating sets

Abstract: The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log Δ) rounds with high probability, where n is the number of nodes, Δ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O (log Δ) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

Tight approximation results for general covering integer programs

Abstract: In this paper we study approximation algorithms for solving a general covering integer program. An n-vector x of nonnegative integers is sought, which minimizes c/sup T//spl middot/x, subject to Ax/spl ges/b, x/spl les/d. The entries of A, b, c are nonnegative. Let m be the number of rows of A. Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, x/spl les/d, on approximately. Two longstanding open questions remain for this general formulation with upper bounds on the variables. (i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the well-known O(log m)-approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same O(log m) multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within O(log m) of the standard LP optimum? (ii) The best known approximation ratio for the problem has been O(log(max/sub j//spl Sigma//sub i/A/sub ij/)) since 1982. This bound can be as bad as polynomial in the input size. Is an O(log m)-approximation, like the one known for the special case of Set Cover, possible? We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smaller-than-integer units. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints.

Design of an optimal continuous speech database for text-to-speech synthesis considered as a set covering problem.

Abstract: Text-to-speech synthesis can be carried out by concatenation of acoustic units obtained from a continuous speech database. This paper presents the optimization of such as database according to phonetic criteria. A large corpus of texts is assembled (311 572 sentences), phonetized automatically and condensed (12 217 sentences) to retain only 10 tokens of the most frequent triphonemes. This is a NP-hard problem of set covering. It has been solved in an approximate way using a greedy algorithm. The condensed database covers 25% of the initial distinct triphonemes, each being represented by 10 tokens at least, which allows 95% of the triphoneme tokens of the initial corpus to be covered. The distribution of the triphonemes remains proportional to their initial statistical appearance.

View planning with a registration constraint

Abstract: The view planning problem, also known as the next-best-view (NBV) problem, for object reconstruction and inspection, has been shown to be isomorphic to the set covering problem which is NP-Complete. In this paper we express a theoretical framework for the NBV problem as an integer programming problem including a registration constraint. Experimental view planning results using a modified greedy search algorithm are presented.

Approximation algorithms for constrained for constrained node weighted steiner tree problems

Abstract: We consider a class of optimization problems, where the input is an undirected graph with two weight functions defined for each node, namely the node's profit and its cost. The goal is to find a connected set of nodes of low cost and high profit. We present approximation algorithms for three natural optimization criteria that arise in this context, all of which are NP-hard. The budget problem asks for maximizing the profit of the set subject to a budget constraint on its cost. The quota problem requires minimizing the cost of the set subject to a quota constraint on its profit. Finally, the prize collecting problem calls for minimizing the cost of the set plus the profit (here interpreted as a penalty) of the complement set. For all three problems, our algorithms give an approximation guarantee of O(\log n), where n is the number of nodes. To the best of our knowledge, these are the first approximation results for the quota problem and for the prize collecting problem, both of which are at least as hard to approximate as set cover. For the budget problem, our results improve on a previous O(\log^2 n) result of Guha, Moss, Naor, and Schieber. Our methods involve new theorems relating tree packings to (node) cut conditions. We also show similar theorems (with better bounds) using edge cut conditions. These imply bounds for the analogous budget and quota problems with edge costs which are comparable to known (constant factor) bounds.

Parallel randomized heuristics for the set covering problem

Abstract: We propose a general scheme to derive heuristics for the Set Covering Problem. The scheme is iterative and embeds constructive heuristics within a randomized procedure. A first group of heuristics is obtained by randomizing the choices made at each step when the solution is constructed in a way similar to that of the so called "Ant System"; a second group of more efficient heuristics is obtained by introducing a random perturbation of the costs of the problem instance. Some computational results are presented. Different parallel implementations are discussed and some performance measures reported.

On applying the set covering model to reseeding

Abstract: The Functional BIST approach is a rather new BIST technique based on exploiting embedded system functionality to generate deterministic test patterns during BIST. The approach takes advantages of two well-known testing techniques, the arithmetic BIST approach and the reseeding method. The main contribution of the present paper consists in formulating the problem of an optimal reseeding computation as an instance of the set covering problem. The proposed approach guarantees high flexibility, is applicable to different functional modules, and, in general, provides a more efficient test set encoding then previous techniques. In addition, the approach shorts the computation time and allows to better exploiting the tradeoff between area overhead and global test length as well as to deal with larger circuits.

Maximally Disjoint Solutions of the Set Covering Problem

Abstract: This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP-complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, extensive computational tests are given demonstrating the nature of these algorithms. These tests are carried out both on randomly generated problems and on problems found in the literature.

Approximation of a Geometric Set Covering Problem

Abstract: We consider a special set covering problem. This problem is a generalization of finding a minimum clique cover in an interval graph. When formulated as an integer program, the 0-1 constraint matrix of this integer program can be partitioned into an interval matrix and a special 0-1 matrix with a single 1 per column. We show that the value of this formulation is bounded by 2k/k+1 times the value of the LP-relaxation, where k is the maximum row sum of the special matrix. For the "smallest" difficult case, i.e., k = 2, this bound is tight. Also we provide an O(n) 3/2 -approximation algorithm in case k = 2.

A Modified Greedy Algorithm for the Set Cover Problem with Weights 1 and 2

Abstract: The set cover problem is that of computing, given a family of weighted subsets of a base set U, a minimum weight subfamily F? such that every element of U is covered by some subset in F?. The k-set cover problem is a variant in which every subset is of size bounded by k. It has been long known that the problem can be approximated within a factor of $$H\left( k \right) = \sum olimits_{i = 1}^k {\left( {{1 \mathord{\left/ {\vphantom {1 i}} \right. \kern- ulldelimiterspace} i}} \right)} $$ by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted k-set cover problem, as a first step towards better approximation of general k- set cover problem, where subset costs are limited to either 1 or 2. It will be shown, via LP duality, that improved approximation bounds of H(3)-1/6 for 3-set cover and H(k)-1/12 for k-set cover can be attained, when the greedy heuristic is suitably modified for this case.

A Modified Greedy Algorithm for the Set Cover Problem with Weights 1 and 2

Abstract: The set cover problem is that of computing, given a family of weighted subsets of a base set U, a minimum weight subfamily F? such that every element of U is covered by some subset in F?. The k-set cover problem is a variant in which every subset is of size bounded by k. It has been long known that the problem can be approximated within a factor of $$H\left( k \right) = \sum olimits_{i = 1}^k {\left( {{1 \mathord{\left/ {\vphantom {1 i}} \right. \kern- ulldelimiterspace} i}} \right)} $$ by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted k-set cover problem, as a first step towards better approximation of general k- set cover problem, where subset costs are limited to either 1 or 2. It will be shown, via LP duality, that improved approximation bounds of H(3)-1/6 for 3-set cover and H(k)-1/12 for k-set cover can be attained, when the greedy heuristic is suitably modified for this case.

Multicasting and broadcasting in undirected WDM networks

Abstract: This paper addresses the problem of multicasting and broadcasting in undirected wavelength-division multiplexing (WDM) networks. Given an undirected network G=(VE) with a source nodes and a set of destination nodes D, /spl Lambda/ is the set of wavelength that can be used in the network. Associated with every edge e, there is a set of available wavelengths on it. The multicast problem is to find a tree rooted at s including all nodes in D such that the cost of the tree is minimum in terms of the cost of wavelength conversion at nodes and the cost of using wavelength on edges. This paper proves that the multicast problem is NP-complete and can not be approximated within a constant factor, unless P=NP. Then we construct an auxiliary graph for the original WDM networks and reduce the multicast problem to a group Steiner tree problem on the auxiliary graph. Employing the known algorithm for the group Steiner tree problem, we derive an algorithm for the problem, which delivers a solution within O(log/sup 2/ (nk)loglog(nk)logp) times the optimum.