

AbstractTraditionally, the theoretical study of NPhard combinatorial optimization problems is based on worstcase analysis of exact solution algorithms, worstcase analysis of polynomialtime approximation algorithms, or averagecase analysis. None of these approaches explain the success of heuristic algorithms in practice. The worstcase analyses are too pessimistic, and averagecase 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 nonempty 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 polynomialtime 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 wellknown 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 EdmondsKarp algorithm for solving the maxflow problem on networks (1971), the landmark paper in complexity theory, "Reducibility Among Combinatorial Problems", in which he proved 21 Problems to be NPcomplete, (1972) the HopcroftKarp 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 RabinKarp string search algorithm (1980). 