SIGEST

Abstract

Theoretical computer science, itself a huge area, has contributed terminology and methodology to many other fields of applied mathematics and computing. Almost everyone in science has heard of the question "Does P = NP?" and of "NP-hard" problems. The SIAM Journal on Computing (SICOMP) represents a wide array of topics in theoretical computer science, the mathematical aspects of computer science. and nonnumerical computing. This issue's SIGEST selection, "A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem," by David R. Karger, first appeared in SICOMP, volume 29 (1999), and has already been honored as one of the three SIAM Paper Prize winners in 2000. This paper is especially appropriate for SIGEST because it not only solves an important, formerly open problem (about which more later) but also presents a concise summary of $\sharp$P-complete problems, polynomial approximation schemes, and fully polynomial randomized approximation schemes.Counting problems, such as "Given a graph, how many distinct Hamiltonian circuits does it have?", involve a determination of the number of solutions, rather than requiring all solutions to be displayed. Such problems arise directly in statistical physics and indirectly in discrete approximations to certain continuous problems. For counting problems, the complexity class $\sharp$P is analogous to the class NP for decision problems. Given that a problem is $\sharp$P-hard, so that efficient algorithms guaranteed to find exact counts are unlikely, a fall-back position is to devise approximation algorithms, techniques guaranteed to find solutions that are provably near-exact, with a precise definition of "near." Development of approximation schemes can be extremely difficult, requiring both algorithm design and complexity-theoretic proofs of lower bounds.The problem addressed in Karger's paper is the classic all-terminal network reliability problem---determining the probability that a network will disconnect when links may fail independently. The paper presents a fully polynomial time approximation scheme that estimates the probability of failure, which is generally harder to approximate, as well as more interesting, than the probability of staying connected (except for very unreliable networks). Even readers unfamiliar with theoretical computer science will be able to appreciate the overall motivation for Karger's approach and the combination of results needed for the proofs. In addition, the paper provides a nice introduction to the strategies of approximation algorithms and to random sampling techniques applied to counting problems in graphs.