Lecture 4 : The Unexplained Success of Heuristic Algorithms for NP-Hard Combinatorial Optimization Problems

By Prof. Richard M. Karp,




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).