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.     

An adaptation of SH heuristic to the location set covering problem

In a recent paper, a new surrogate heuristic (SH) has been proposed for the set covering problem. Here we present an adaptation of it in order to solve more efficiently the location set covering problem. We will show that our new version not only outperforms algorithm SH but that it is more accurate than the pair CMA/FMC. Its power is experimentally tested over a set of 65 randomly generated problems.     

Surrogate constraint normalization for the set covering problem

The set covering problem (SCP) is central in a wide variety of practical applications for which finding good feasible solutions quickly (often in real-time) is crucial. Surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming. This framework remains the core of the most effective constructive heuristics for the solution of the SCP chiefly represented by the widely-used Chvátal method. This paper introduces a number of normalization rules and demonstrates their superiority to the classical Chvátal rule, especially when solving large scale and real-world instances. Directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided.     

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.     

MIP reformulations of the probabilistic set covering problem

In this paper, we address the following probabilistic version (PSC) of the set covering problem: where A is a 0-1 matrix, is a random 0-1 vector and is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of are pairwise independent, the distribution function of is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second. Anureet Saxena’s research was supported by the National Science Foundation through grant #DMI-0352885 and by the Office of Naval Research through contract N00014-03-1-0133. Vineet Goyal’s research was supported in part by NSF grant CCF-0430751 and ITR grant CCR-0122581.     

A Lagrangian-based heuristic for large-scale set covering problems

We present a new Lagrangian-based heuristic for solving large-scale set-covering problems arising from crew-scheduling at the Italian Railways (Ferrovie dello Stato). Our heuristic obtained impressive results when compared to state-of-the-art codes on a test-bed provided by the company, which includes instances with sizes ranging from 50,000 variables and 500 constraints to 1,000,000 variables and 5000 constraints. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.This research was performed while the author was affiliated with IASI, CNR and Dipartimento di Informatica e Sistemistica, Università di Roma, La Sapienza, Italy.This research was partially supported by National Research Program Metodi di Ottimizzazione per le Decisioni, MURST, Roma, Italy.     

A note on a symmetrical set covering problem: The lottery problem

In a lottery, n numbers are drawn from a set of m numbers. On a lottery ticket we fill out n numbers. Consider the following problem: what is the minimum number of tickets so that there is at least one ticket with at least p matching numbers? We provide a set-covering formulation for this problem and characterize its LP solution. The existence of many symmetrical alternative solutions, makes this a very difficult problem to solve, as our computational results indicate.     

A theoretical justification of the set covering greedy heuristic of Caprara et al.

Large scale set covering problems have often been approached by constructive greedy heuristics, and much research has been devoted to the design and evaluation of various greedy criteria for such heuristics. A criterion proposed by Caprara et al. (1999) is based on reduced costs with respect to the yet unfulfilled constraints, and the resulting greedy heuristic is reported to be superior to those based on original costs or ordinary reduced costs.We give a theoretical justification of the greedy criterion proposed by Caprara et al. by deriving it from a global optimality condition for general non-convex optimisation problems. It is shown that this criterion is in fact greedy with respect to incremental contributions to a quantity which at termination coincides with the deviation between a Lagrangian dual bound and the objective value of the feasible solution found.     

A Lagrangean heuristic for the maximal covering location problem

We develop a Lagrangean heuristic for the maximal covering location problem. Upper bounds are given by a vertex addition and substitution heuristic and lower bounds are produced through a subgradient optimization algorithm. The procedure was tested in networks of up to 150 vertices. A duality gap was generally present at the end of the heuristic for the larger problems. The test problems were run in an IBM 3090-600J ‘super-computer’; the maximum computing time was kept below three minutes of CPU.     

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.     

An iterated-tabu-search heuristic for a variant of the partial set covering problem

In this paper, we propose a heuristic algorithm to solve a new variant of the partial set covering problem. In this variant, each element $e_i$ has a gain $g_i$ (i.e., a positive profit), each set $s_j$ has a cost $c_j$ (i.e., a negative profit), and each set $s_j$ is part of a unique group $G_k$ that has a fixed cost $f_k$ (i.e., a negative profit). The objective is to maximize profit and it is not necessary to cover all of the elements. We present an industrial application of the model and propose a hybrid heuristic algorithm to solve it; the proposed algorithm is an iterated-local-search algorithm that uses two levels of perturbations and a tabu-search heuristic. Whereas the first level of perturbation diversifies the search around the current local optimum, the second level of perturbation performs long jumps in the search space to help escape from local optima with large basins of attraction. The proposed algorithm is evaluated on thirty real-world problems and compared to a memetic algorithm. Computational results show that most of the solutions found by ITS are either optimal or very close to optimality.     

Column generation extensions of set covering greedy heuristics

Large-scale set covering problems are often approached by constructive greedy heuristics, and many selection criteria for such heuristics have been considered. These criteria are typically based on measures of the cost of setting an additional variable to one in relation to the number of yet unfulfilled constraints that it will satisfy. We show how such greedy selections can be performed on column-oriented set covering models, by using a fractional optimization formulation and solving sequences of ordinary column generation problems for the application at hand.     

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.     

An aggressive reduction scheme for the simple plant location problem

Pisinger et al. introduced the concept of ‘aggressive reduction’ for large-scale combinatorial optimization problems. The idea is to spend much time and effort in reducing the size of the instance, in the hope that the reduced instance will then be small enough to be solved by an exact algorithm.     

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.     

A probabilistic heuristic for a computationally difficult set covering problem

An efficient probabilistic set covering heuristic is presented. The heuristic is evaluated on empirically difficult to solve set covering problems that arise from Steiner triple systems. The optimal solution to only a few of these instances is known. The heuristic provides these solutions as well as the best known solutions to all other instances attempted.     

An algorithm for set covering problem

In this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent, subgradient optimisation and linear programming. Computational results are presented for problems involving up to 400 rows and 4000 columns.     

A hybrid heuristic for the maximum clique problem

In this paper we present a heuristic based steady-state genetic algorithm for the maximum clique problem. The steady-state genetic algorithm generates cliques, which are then extended into maximal cliques by the heuristic. We compare our algorithm with three best evolutionary approaches and the overall best approach, which is non-evolutionary, for the maximum clique problem and find that our algorithm outperforms all the three evolutionary approaches in terms of best and average clique sizes found on majority of DIMACS benchmark instances. However, the obtained results are much inferior to those obtained with the best approach for the maximum clique problem.