Abstract
Traditionally, the theoretical study of NP-hard combinatorial optimization problems is based on worst-case analysis of exact solution algorithms, worst-case analysis of polynomial-time approximation algorithms, or average-case analysis. None of these approaches explain the success of heuristic algorithms in practice. The worst-case analyses are too pessimistic, and average-case analysis requires unjustified probabilistic assumptions.We will suggest an alternative, more empirical, approach.
To illustrate our approach we introduce the class of implicit hitting set problems. A hitting set problem is specified by a finite ground set U, a weight w(x) for each element x in U, and a family of subsets of U called circuits. A hitting set is defined as a subset of U having a non-empty intersection with each circuit.The problem is to find a hitting set of minimum weight.
An implicit hitting set problem is one in which the set of circuits is not listed explicitly but instead is specified by a separation oracle: a polynomial-time algorithm which, given a subset H of the ground set, either certifies that H is a hitting set or returns a circuit that is not hit bv H.
We shall exhibit several well-known problems that can be cast as implicit hitting set problems, give a generic heuristic algorithm for solving implicit hitting set problems, and describe our computational experience with a particular implicit hitting set problem involving the global multiple alignment of several genomes. This is joint work with Erick Moreneo Centeno.
Biodata of the Speaker
Richard M. Karp is among the most eminent computer scientists in the world, notable for his research in the theory of algorithms, for which he received a Turing Award in 1985, The Benjamin Franklin Medal in Computer and Cognitive Science in 2004, and the Kyoto Prize in 2008. He is a Fellow of the Association for Computing Machinery and is the recipient of several honorary degrees. Among many influential works, some of his prominent discoveries include, the Edmonds-Karp algorithm for solving the max-flow problem on networks (1971), the landmark paper in complexity theory, "Reducibility Among Combinatorial Problems", in which he proved 21 Problems to be NP-complete, (1972) the Hopcroft-Karp algorithm for finding maximum cardinality matchings in bipartite graph (1973), the Karp Lipton theorem (which proves that if SAT can be solved by Boolean circuits with a polynomial number of logic gates, then the polynomial hierarchy collapses to its second level) and the Rabin-Karp string search algorithm (1980).