A Two-Phase Heuristic for Set Covering

The set covering problem (SCP) is a well-known computationally intractable problem. We suggest here a two-phase heuristic to solve it. The first phase reduces substantially the size of the given SCP by removing some variables; the second phase applies a simple Lagrangian heuristic applied to the reduced problem. Construction and improvement heuristics are embedded in the Lagrangian solution approach. The construction heuristic provides good covers by solving small SCPs. The improvement heuristic inserts these covers into larger ones from which better covers are extracted, again by solving different but also small SCPs. The novelty lies in the reduction of the problem size by an effective variable-fixing heuristic, which, in practice, eliminates up to 95% of the variables of the problem without sacrificing the solution quality. Extensive computational and comparative results are presented

Statistical mechanics of the vertex-cover problem

We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC exhibits an coverable-uncoverable phase transition. Very close to this transition, depending on the solution algorithm, easy-hard transitions in the typical running time of the algorithms occur. We explain a statistical mechanics approach, which works by mapping VC to a hard-core lattice gas, and then applying techniques like the replica trick or the cavity approach. Using these methods, the phase diagram of VC could be obtained exactly for connectivities $c<e$, where VC is replica symmetric. Recently, this result could be confirmed using traditional mathematical techniques. For $c>e$, the solution of VC exhibits full replica symmetry breaking. The statistical mechanics approach can also be used to study analytically the typical running time of simple complete and incomplete algorithms for VC. Finally, we describe recent results for VC when studied on other ensembles of finite- and infinite-dimensional graphs.Comment: review article, 26 pages, 9 figures, to appear in J. Phys. A: Math. Ge

Data-Collection for the Sloan Digital Sky Survey: a Network-Flow Heuristic

The goal of the Sloan Digital Sky Survey is ``to map in detail one-quarter of the entire sky, determining the positions and absolute brightnesses of more than 100 million celestial objects''. The survey will be performed by taking ``snapshots'' through a large telescope. Each snapshot can capture up to 600 objects from a small circle of the sky. This paper describes the design and implementation of the algorithm that is being used to determine the snapshots so as to minimize their number. The problem is NP-hard in general; the algorithm described is a heuristic, based on Lagriangian-relaxation and min-cost network flow. It gets within 5-15% of a naive lower bound, whereas using a ``uniform'' cover only gets within 25-35%.Comment: proceedings version appeared in ACM-SIAM Symposium on Discrete Algorithms (1998

A regret model applied to the maximum capture location problem

This article addresses issues related to location and allocation problems. Herein, we intend to demonstrate the influence of congestion, through the random number generation, of such systems in final solutions. An algorithm is presented which, in addition to the GRASP, incorporates the Regret with the pminmax method to evaluate the heuristic solution obtained with regard to its robustness for different scenarios. Taking as our point of departure the Maximum Capture Location Problem proposed by Church and Revelle [1, 26], an alternative perspective is added in which the choice behavior of the server does not depend only on the elapsed time from the demand point looking to the center, but includes also the service waiting time.N/

On the design of custom packs: grouping of medical disposable items for surgeries

A custom pack combines medical disposable items into a single sterile package that is used for surgical procedures. Although custom packs are gaining importance in hospitals due to their potential benefits in reducing surgery setup times, little is known on methodologies to configure them, especially if the number of medical items, procedure types and surgeons is large. In this paper, we propose a mathematical programming approach to guide hospitals in developing or reconfiguring their custom packs. In particular, we are interested in minimising points of touch, which we define as a measure for physical contact between staff and medical materials. Starting from an integer non-linear programming model, we develop both an exact linear programming (LP) solution approach and an LP-based heuristic. Next, we also describe a simulated annealing approach to benchmark the mathematical programming methods. A computational experiment, based on real data of a medium-sized Belgian hospital, compares the optimised results with the performance of the hospital’s current configuration settings and indicates how to improve future usage. Next to this base case, we introduce scenarios in which we examine to what extent the results are sensitive for waste, i.e. adding more items to the custom pack than is technically required for some of the custom pack’s procedures, since this can increase its applicability towards other procedures. We point at some interesting insights that can be taken up by the hospital management to guide the configuration and accompanying negotiation processes

A regret model applied to the facility location problem with limited capacity facilities

This article addresses issues related to location and allocation problems. Herein, we intend to demonstrate the influence of congestion, through the random number generation, of such systems in final solutions. An algorithm is presented which, in addition to the GRASP, incorporates the Regret with the pminmax method to evaluate the heuristic solution obtained with regard to its robustness for different scenarios. Taking as our point of departure the Facility Location Problem proposed by Balinski [27], an alternative perspective is added associating regret values to particular solutions.N/

Efficient state reduction methods for PLA-based sequential circuits

Experiences with heuristics for the state reduction of finite-state machines are presented and two new heuristic algorithms described in detail. Results on machines from the literature and from the MCNC benchmark set are shown. The area of the PLA implementation of the combinational component and the design time are used as figures of merit. The comparison of such parameters, when the state reduction step is included in the design process and when it is not, suggests that fast state-reduction heuristics should be implemented within FSM automatic synthesis systems

A regret model applied to the maximum coverage location problem with queue discipline

This article discusses issues related to the location and allocation problems where is intended to demonstrate, through the random number generation, the influence of congestion of such systems in the final solutions. It is presented an algorithm that, in addition to the GRASP, incorporates the Regret with the pminmax method to evaluate the heuristic solution obtained in regard to its robustness for different scenarios. To the well know Maximum Coverage Location Problem from Church and Revelle [1] an alternative perspective is added in which the choice behavior of the server does not only depend on the elapsed time from the demand point looking to the center, but also includes the waiting time for service conditioned by a waiting queue.N/