Showing papers on "Set cover problem published in 2003"

Strategyproof cost-sharing mechanisms for set cover and facility location games

Abstract: Strategyproof cost-sharing mechanisms, lying in the core, that recover 1/@a fraction of the cost, are presented for the set cover and facility location games: @a=O(log n) for the former and 1:861 for the latter. Our mechanisms utilize approximation algorithms for these problems based on the method of dual-fitting.

Solving the set cover problem and the problem of exact cover by 3-sets in the Adleman-Lipton model.

Abstract: Adleman wrote the first paper in which it is shown that deoxyribonucleic acid (DNA) strands could be employed towards calculating solutions to an instance of the NP-complete Hamiltonian path problem (HPP). Lipton also demonstrated that Adleman's techniques could be used to solve the NP-complete satisfiability (SAT) problem (the first NP-complete problem). In this paper, it is proved how the DNA operations presented by Adleman and Lipton can be used for developing DNA algorithms to resolving the set cover problem and the problem of exact cover by 3-sets.

The Multi-level Location Set Covering Model

Abstract: The classical Location Set Covering Problem involves finding the smallest number of facilities and their locations so that each demand is covered by at least one facility. It was first introduced by Toregas in 1970. This problem can represent several different application settings including the location of emergency services and the selection of conservation sites. The Location Set Covering Problem can be formulated as a 0-1 integer- programming model. Roth (1969) and Toregas and ReVelle (1973) developed reduction approaches that can systematically eliminate redundant columns and rows as well as identify essential sites. Such approaches can often reduce a problem to a size that is considerably smaller and easily solved by linear programming using branch and bound. Extensions to the Location Set Covering Model have been proposed so that additional levels of coverage are either encouraged or required. This paper focuses on one of the extended model forms called the Multi-level Location Set Covering Model. The reduction rules of Roth and of Toregas and ReVelle violate properties found in the multi-level model. This paper proposes a new set of reduction rules that can be used for the multi-level model as well as the classic single-level model. A demonstration of these new reduction rules is presented which indicates that such problems may be subject to significant reductions in both the numbers of demands as well as sites.

An improved approximation algorithm for vertex cover with hard capacities

Abstract: In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G = (V,E), the goal is to cover all the edges by picking a minimum cover using the vertices. When we pick a vertex, we can cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. Previously, 2-approximation algorithms were developed with the assumption that multiple copies of a vertex may be chosen in the cover. If we are allowed to pick at most a given number of copies of each vertex, then the problem is significantly harder to solve. Chuzhoy and Naor (Proc. IEEE Symposium on Foundations of Computer Science, 481-489, 2002 ) have recently shown that the weighted version of this problem is at least as hard as set cover; they have also developed a 3-approximation algorithm for the unweighted version. We give a 2-approximation algorithm for the unweighted version, improving the Chuzhoy-Naor bound of 3 and matching (up to lower-order terms) the best approximation ratio known for the vertex cover problem.

Efficient sequences of trials

Abstract: We introduce a problem called sequential trial optimization, a generalization of the well studied set cover problem with a new objective function. We give a simple algorithm that achieves a constant factor approximation to this problem. Sequential trial optimization naturally arises in heterogenous search environments such as peer to peer networks.

Solution Techniques for the Large Set Covering Problem

Abstract: Given a finite set E and a family F={E"1,...,E"m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E"i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.

Scheduling of train driver for taiwan railway administration

Abstract: The paper presents the experience of two approaches in solving driver-scheduling problem for a depot at Taiwan Railway Administration. The driver-scheduling problem consists of generating feasible duties, selecting a schedule of duties for the depot, and circulating the duties in the schedule into a roster for each driver. By using mathematical programming approach, the set covering problem and the constrained traveling salesman problem are solved for the pairing and rostering problems respectively. Comparing to TRA’s practical solution, we find a schedule with less duties, but a roster with a little longer cycle length. By using Genetic algorithm approach, we have a very flexible process to deal with all kinds of rules and objectives in the pairing and rostering problems. In the study, we consider multiple objectives for both pairing and rostering problems, and obtain very good solutions.

A computational study of a genetic algorithm to solve the set covering problem

Abstract: A Genetic Algorithm to solve the Set Covering Problem has been proposed by K. Iwamura, T. Sibahara, M. Fushimi and H. Morohoshi [14, 15]. In their algorithm, they have made some improvements in getting some better feasible solutions, i.e. better chromosomes at the first starting population, taking full account of Domain Specific Knowledge with sound programming skill. Here, we have carried out a systematic computational study of the algorithm to show its efficiency. We show computing time dependency on problem size, parameters of the Genetic Algorithm. And finally we make a computational comparison between the Genetic Algorithm and the commercial software code LINGO4 concerning approximation ratio and computing time.

Multicasting to groups in optical networks and related combinatorial optimization problems

Abstract: Given a source node and a family D of subsets of nodes of a WDM optical network, the multicasting to groups (MG) problem is to find a set of paths from the source to at least one node in each subset in D, and an assignment of wavelengths to paths so that paths sharing an optical link are assigned different wavelengths. The goal is to minimize the total number of used wavelengths. We note that MG is closely related to several important combinatorial optimization problems. These include set cover and some useful generalizations of it, that correspond to MG when the network is a tree. From the equivalence between MG and set cover it follows that unless NP /spl sub/ DTIME(n/sup log log/ /sup n/), MG cannot be approximated within a logarithmic factor. On the positive side, we give polynomial time approximation algorithms for the MG problem. Our algorithm has a guaranteed approximation factor matching the non-approximability bound in case of tree networks.

Covering with Ellipses

Abstract: We address the problem of how to cover a set of required points by a small number of axis-parallel ellipses that avoid a second set of forbidden points. We study geometric properties of such covers and present an efficient randomized approximation algorithm for the cover construction. This question is motivated by a special pattern recognition task where one has to identify ellipse-shaped protein spots in two-dimensional electrophoresis images.

Optimal k-needle placement strategy considering an approximate initial needle position

Abstract: Techniques are provided to determine, for a given set of possible initial needle positions, the smallest set of needles needed to guarantee a successful biopsy. Advantageously, this problem may be formulated as a “Set Covering Problem” (SCP), a well-known combinatorial optimization problem for which good approximations are known. Additionally, the present invention provides techniques to maximize the coverage of the possible initial positions for a given maximum number of k needles. This aspect of the invention may be formulated as a “Maximum k-Coverage Problem”.

MRPF: An Architectural Transformation for Synthesis of High-Performance and Low-Power Digital Filters

Abstract: We present a graph theoretical methodology that reduces the implementation complexity of a vector multiplied by a scalar The proposed approach is called MRP (minimally redundant parallel) optimization and is presented in FIR filtering framework to obtain a low-complexity multiplierless implementation. The key idea is to expand the design space using shift inclusive differential coefficients together with computation reordering using a graph theoretic approach to obtain maximal computation sharing. The transformed architecture of a filter is obtained by solving a set cover problem of the graph. A simple algorithm based on a greedy approach is presented. The proposed approach is merged with common sub-expression elimination. The simulation results show that 70% and 16% improvement in terms of computational complexity over simple implementation (transposed direct form) and common sub-expression, respectively, when using carry, lookahead adder synthesized from Synopsys designware library in 0.25 /spl mu/m technology.

An improved approximation algorithm for vertex cover with hard capacities (extended abstract)

Abstract: In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G =( V, E), the goal is to cover all the edges by picking a minimum cover using the vertices. When we pick a vertex, we can cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. Previously, 2-approximation algorithms were developed with the assumption that multiple copies of a vertex may be chosen in the cover. If we are allowed to pick at most a given number of copies of each vertex, then the problem is significantly harder to solve. Chuzhoy and Naor (Proc. IEEE Symposium on Foundations of Computer Science, 481-489, 2002 ) have recently shown that the weighted version of this problem is at least as hard as set cover; they have also developed a 3-approximation algorithm for the unweighted version. We give a 2-approximation algorithm for the unweighted version, improv- ing the Chuzhoy-Naor bound of 3 and matching (up to lower-order terms) the best approximation ratio known for the vertex cover problem.

Set Cover via Dual Fitting

Abstract: In this chapter we will introduce the method of dual fitting, which helps analyze combinatorial algorithms using LP-duality theory. Using this method, we will present an alternative analysis of the natural greedy algorithm (Algorithm 2.2) for the set cover problem (Problem 2.1). Recall that in Section 2.1 we deferred giving the lower bounding method on which this algorithm was based. We will provide the answer below. The power of this approach will become apparent when we show the ease with which it extends to solving several generalizations of the set cover problem (see Section 13.2).

Lagrangean price directive ring generation for network design

Abstract: When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.

A ring generation problem based on the traveling salesman subtour problem

Abstract: When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.

A ring network design problem solved by a ring generation and allocation approach

Abstract: When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.

Rounding Applied to Set Cover

Abstract: We will introduce the technique of LP-rounding by using it to design two approximation algorithms for the set cover problem, Problem 2.1. The first is a simple rounding algorithm achieving a guarantee of f, where f is the frequency of the most frequent element. The second algorithm, achieving an approximation guarantee of O(log n), illustrates the use of randomization in rounding.

Input Data Dependency of a Genetic Algorithm to Solve the Set Covering Problem

Abstract: A genetic algorithm to solve the set covering problem proposed in the literature had some improvements which gave better solutions, i.e., better chromosomes in the first starting population, taking full account of domain specific knowledge with sound programming skill. We have further investigated the input data dependency of their genetic algorithm, i.e., the dependency on costs and density. We have found that for input problem data sets with densities greater than or equal to 3%, our genetic algorithm is still practical both in computing time and approximation ratio.

Calculating cost coefficients for generation of rings in network design

Abstract: When designing a telecommunication network, one often wish to include some kind of survivability requirement, for example that the network should be two-connected. A two-connected network fulfills the requirement that there should be at least two paths with no links in common between all pairs of nodes. One form of design model is to prescribe that the network should be composed of connected rings of links. The network design problem is then to choose links from a given network, and compose them into a number of rings. A ring is reliable in the sense that there always exist two ways of sending traffic, clockwise or counter-clockwise, which means that a ring fulfills the two-connectivity requirement. There is often a number of requirements on a ring, such as a limited length and limited number of nodes connected to the ring. This means that a ring network will include a number of rings, and traffic between rings must be possible. The traffic between rings is usually made at certain nodes, called transit nodes. Therefore all rings should be connected to at least one of the transit nodes. We focus on the case where we have two transit nodes in the network.Each possible ring is associated with a certain fixed cost, and all links in a certain ring are given the same capacity. Reserve capacity is allocated according to certain principles. The number of possible rings in a network is an exponential function of the number of nodes in the network, so for larger networks is it impossible to a priori generate all possible rings.We describe the problem, and model it as a linear integer programming problem, where a set of rings are assumed to be known. The usage of rings, i.e., the allocation of demand to rings, is determined. In practice, too many rings can not be included in the model. Instead we must be able to generate useful rings. A Lagrangean relaxation of the model is formulated, and the dual solution is used in order to derive reduced costs which can be used to generate new better rings. The information generated describes only the physical structure of the ring, not the usage of it. The ring generation problem is a modified traveling salesman subtour problem, which is known to be difficult to solve. Therefore, we focus on heuristic solution methods for this problem.We also presents a column generation approach where the problem is modeled as a set covering problem. Here, a column describes both the topology of the ring and the exact usage of it. A similar ring generation problem appears as a subproblem, in order to generate new rings.All methods are computationally tested on both real life data and randomly generated data, similar to real life problems.

Set Cover via the Primal-Dual Schema

Abstract: As noted in Section 12.3, the primal-dual schema is the method of choice for designing approximation algorithms since it yields combinatorial algorithms with good approximation factors and good running times. We will first present the central ideas behind this schema and then use it to design a simple f factor algorithm for set cover, where f is the frequency of the most frequent element.