Two new heuristics for the location set covering problem

Summary In this paper we present two new greedy-type heuristics for solving the location set covering problem. We compare our new pair of algorithms with the pair GH1 and GH2 [Vasko and Wilson (1986)] and show that they perform better for a selected set of test problems.     

A probabilistic analysis of the set covering problem

A probabilistic analysis of the minimum cardinality set covering problem (SCP) is developed, considering a stochastic model of the (SCP), withn variables andm constraints, in which the entries of the corresponding (m, n) incidence matrix are independent Bernoulli distributed random variables, each with constant probabilityp of success. The behaviour of the optimal solution of the (SCP) is then investigated as bothm andn grow asymptotically large, assuming either an incremental model for the evolution of the matrix (for each size, the matrixA is obtained bordering a matrix of smaller size by new columns and rows) or an independent one (for each size, an entirely new set of entries forA are considered). Two functions ofm are identified, which represent a lower and an upper bound onn in order the (SCP) to be a.e. feasible and not trivial. Then, forn lying within these bounds, an asymptotic formula for the optimum value of the (SCP) is derived and shown to hold a.e.The performance of two simple randomized algorithms is then analyzed. It is shown that one of them produces a solution value whose ratio to the optimum value asymptotically approaches 1 a.e. in the incremental model, but not in the independent one, in which case the ratio is proved to be tightly bounded by 2 a.e. Thus, in order to improve the above result, a second randomized algorithm is proposed, for which it is proved that the ratio between the approximate solution value and the optimum approaches 1 a.e. also in the independent model.     

Facets and lifting procedures for the set covering polytope

Given a family of subsets of an arbitrary groundsetE, acover of is any setC E having non-empty intersection with every subset in. In this paper we deal with thecovering polytope, i.e., the convex hull of the incidence vectors of all the covers of. In Section 2 we review all the known properties of the covering polytope. In Sections 3 and 4 we introduce two new classes of non-Boolean facets of such a polytope. In Sections 5 and 6 we describe some non-sequential lifting procedures. In Section 7 a generalization of the notion ofweb introduced by L.E. Trotter is presented together with the facets of the covering polytope produced by such a structure.Moreover, the strong connections between several combinatorial problems and the covering problem are pointed out and, exploiting those connections, some examples are presented of new facets for the Knapsack and Acyclic Subdigraph polytopes.     

An electromagnetism metaheuristic for the unicost set covering problem

In this paper we propose a new heuristic algorithm to solve the unicost version of the well-known set covering problem. The method is based on the electromagnetism metaheuristic approach which, after generating a pool of solutions to create the initial population, applies a fixed number of local search and movement iterations based on the “electromagnetism” theory. In addition to some random aspects, used in the construction and local search phases, we also apply mutation in order to further escape from local optima.     

On the 0, 1 facets of the set covering polytope

In this paper, we consider inequalities of the form _jx_j , where _j equals 0 or 1, and is a positive integer. We give necessary and sufficient conditions for such inequalities to define facets of the set covering polytope associated with a 0, 1 constraint matrixA. These conditions are in terms of critical edges and critical cutsets defined in the bipartite incidence graph ofA, and are in the spirit of the work of Balas and Zemel on the set packing problem where similar notions were defined in the intersection graph ofA. Furthermore, we give a polynomial characterization of a class of 0, 1 facets defined from chorded cycles of the bipartite incidence graph. This characterization also yields all the 0, 1 liftings of odd-hole inequalities for the simple plant location polytope.Research partially supported by NSF grant ECS-8601660 and AFORS grant 87-0292.     

Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean -space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

Clustering heuristics for set covering

We introduce a new class of set covering heuristics, based on clustering techniques. In its simplest form, a heuristic in this class may be described as follows: firstly, partition the column set into clusters formed by columns that are close to each other (e.g. in the Hamming distance sense). Then select a best (e.g. a cheapest) column in each cluster; if the selected columns form a coverC, then extract fromC a prime cover and stop; else, modify the partition (e.g. by increasing the number of clusters) and repeat. We describe two implementations of this general algorithmic strategy, relying on the Single Linkage and the Leader clustering algorithm, respectively. Numerical experiments performed on 72 randomly generated test problems with 200 or 400 rows and 1000 columns indicate that the above two heuristics often yield cheaper covers than other well-known (greedy-type) heuristics when the cost-range is not too narrow.The present work is based on R.K. Kwatera's dissertation, written under the supervision of B. Simeone. A preliminary version was presented at EURO VIII, Paris, July 1988.     

一类最小覆盖问题

对于任意给定的平行四边形,确定了与其相似的直径为1的三角形集族的最小平行四边形覆盖.     

An interior point algorithm to solve computationally difficult set covering problems

We present an interior point approach to the zero–one integer programming feasibility problem based on the minimization of a nonconvex potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems.     

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.     

Distributionally scrambled set and minimal set

We investigate the relation between distributional chaos and minimal sets, and discuss how to obtain various distributionally scrambled sets by using least and simplest minimal sets. We show: i) an uncountable extremal distributionally scrambled set can appear in a system with just one simple minimal set: a periodic orbit with period 2; ii) an uncountable dense invariant distributionally scrambled set can occur in a system with just two minimal sets: a fixed point and an infinite minimal set; iii) infinitely many minimal sets are necessary to generate a uniform invariant distributionally scrambled set, and an uncountable dense extremal invariant distributionally scrambled set can be constructed by using just countably infinitely many periodic orbits.     

A surprising covering of the real line

We construct an increasing sequence of Borel subsets of , such that their union is , but cannot be covered with countably many translations of one set. The proof uses a random method.

     

Dominating sets with small clique covering number

Motivated by earlier work on dominating cliques, we show that if a graph G is connected and contains no induced subgraph isomorphic to P₆ or H_t (the graph obtained by subdividing each edge of K_1,t, t ≥ 3, by exactly one vertex), then G has a dominating set which induces a connected graph with clique covering number at most t − 1. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 101–105, 1997     

An effective and simple heuristic for the set covering problem

This paper investigates the development of an effective heuristic to solve the set covering problem (SCP) by applying the meta-heuristic Meta-RaPS (Meta-heuristic for Randomized Priority Search). In Meta-RaPS, a feasible solution is generated by introducing random factors into a construction method. Then the feasible solutions can be improved by an improvement heuristic. In addition to applying the basic Meta-RaPS, the heuristic developed herein integrates the elements of randomizing the selection of priority rules, penalizing the worst columns when the searching space is highly condensed, and defining the core problem to speedup the algorithm. This heuristic has been tested on 80 SCP instances from the OR-Library. The sizes of the problems are up to 1000 rows × 10,000 columns for non-unicost SCP, and 28,160 rows × 11,264 columns for the unicost SCP. This heuristic is only one of two known SCP heuristics to find all optimal/best known solutions for those non-unicost instances. In addition, this heuristic is the best for unicost problems among the heuristics in terms of solution quality. Furthermore, evolving from a simple greedy heuristic, it is simple and easy to code. This heuristic enriches the options of practitioners in the optimization area.     

An axiomatization of minimal curb sets

Norde et al. [Games Econ. Behav. 12 (1996) 219] proved that none of the equilibrium concepts in the literature on equilibrium selection in finite strategic games satisfying existence is consistent. A transition to set-valued solution concepts overcomes the inconsistency problem: there is a multiplicity of consistent set-valued solution concepts that satisfy nonemptiness and recommend utility maximization in one-player games. The minimal curb sets of Basu and Weibull [Econ. Letters 36 (1991) 141] constitute one such solution concept; this solution concept is axiomatized in this article.