Combinational optimization problems for which almost every algorithm is asymptotically optimal

Abstract

Consider a class of optimization problems for which the cardinality of the set of feasible solutions is m and the size of every feasible solution is N. We prove in a general probabilistic framework that the value of the optimal solution and the value of the worst solution are asymptotically almost surely (a.s.) the same provided as N and m become large. This result implies that for such a class of combinatorial optimization problems almost ecery algorithm finds asymptotically optimal solution! The quadratic assignment problem, a location problem on graphs, and a pattern matching problem fall into this class

Keywords:

• Combinatorial optimization
• Asymptotically optimal solution
• Quadratic assignment problem
• Location problem on graphs
• Pattern matching problem

$sub:∗$esub:This research was primary done while the author was visiting INRIA Rocquencourt, France, and he wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for a generous support. In addition, support was provided by NSF Grants CCR-9201078, NCR-9206315 and INT-8912631, by Grant AFOSR-90-0107, and in part by NATO Collaborative Grant 0057 89

$sub:∗$esub:This research was primary done while the author was visiting INRIA Rocquencourt, France, and he wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for a generous support. In addition, support was provided by NSF Grants CCR-9201078, NCR-9206315 and INT-8912631, by Grant AFOSR-90-0107, and in part by NATO Collaborative Grant 0057 89

Notes

$sub:∗$esub:This research was primary done while the author was visiting INRIA Rocquencourt, France, and he wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for a generous support. In addition, support was provided by NSF Grants CCR-9201078, NCR-9206315 and INT-8912631, by Grant AFOSR-90-0107, and in part by NATO Collaborative Grant 0057 89