The multicovering problem

Abstract The multicovering problem is: Min CX subject to AX ⩾ b, X ϵ {0,}, where A is a matrix whose elements are all zero or one and b is a vector of positive integers: this problem has many applications in scheduling, location and business areas. A variety of industrial, commercial and military contexts give rise to large problems of this kind, frequently at or beyond the limits of current computational tractability. Our research goal is the development of a special-purpose algorithm capable of solving large problems of this kind. Conceptual and computational inputs to the algorithm arise from three sources: from the generalization of computational tools developed for more specific problems (in particular the set covering problem), from the adaptation of general integer programming and combinatorial optimization methods, and from the development within this paper of special techniques designed to take advantage of the structure of the problem studied. Following a description of the algorithm we give some computational results for randomly generated problems.

[1]  Adi Ben-Israel,et al.  Vector-Valued Criteria in Geometric Programming , 1971, Oper. Res..

[2]  Vasek Chvátal,et al.  A Greedy Heuristic for the Set-Covering Problem , 1979, Math. Oper. Res..

[3]  E. Balas.  Cutting planes from conditional bounds: A new approach to set covering , 1980 .

[4]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[5]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[6]  E. Balas.  A class of location, distribution and scheduling problems : modeling and solution methods , 1982 .

[7]  J. Beasley.  An algorithm for set covering problem , 1987 .

[8]  Dorit S. Hochbaum,et al.  Database Location in Computer Networks , 1980, JACM.

[9]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[10]  G. Nemhauser,et al.  Exceptional Paper—Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms , 1977 .

[11]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[12]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[13]  Javier Etcheberry,et al.  The Set-Covering Problem: A New Implicit Enumeration Algorithm , 1977, Oper. Res..

[14]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[15]  Nicos Christofides,et al.  Algorithms for large scale set covering problems , 1993, Ann. Oper. Res..

[16]  Kurt Spielberg,et al.  Set Covering by Single-Branch Enumeration with Linear-Programming Subproblems , 1971, Oper. Res..

[17]  H. Crowder,et al.  Cluster Analysis: An Application of Lagrangian Relaxation , 1979 .

[18]  Andrew C. Ho,et al.  Set covering algorithms using cutting planes, heuristics, and subgradient optimization: A computational study , 1980 .

[19]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[20]  Dorit S. Hochbaum,et al.  Approximation Algorithms for the Set Covering and Vertex Cover Problems , 1982, SIAM J. Comput..

[21]  Gregory Dobson,et al.  Worst-Case Analysis of Greedy Heuristics for Integer Programming with Nonnegative Data , 1982, Math. Oper. Res..

[22]  Dorit S. Hochbaum,et al.  A fast approximation algorithm for the multicovering problem , 1986, Discret. Appl. Math..

[23]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[24]  Angela Margaret Hey.  Algorithms for the set covering problem , 1981 .

[25]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..