Michael Luby

Monte-Carlo algorithms for enumeration and reliability problems

1. Introduction We present a simple but very general Monte-Carlo technique for the approximate solution of enumeration and reliability problems. Several applications are given, including: 1. Estimating the number of triangulated plane maps with a given number of ver-tices; 2. Estimating the cardinality of a union of sets; 3. Estimating the number of input combinations for which a boolean function, presented in disjunctive normal form,

Monte-Carlo approximation algorithms for enumeration problems

We develop polynomial time Monte-Carlo algorithms which produce good approximate solutions to enumeration problems for which it is known that the computation of the exact solution is very hard. We start by developing a Monte-Carlo approximation algorithm for the DNF counting problem, which is the problem of counting the number of satisfying truth assignments to a formula in disjunctive normal form. The input to the algorithm is the formula and two parameters e and δ. The algorithm produces an estimate which is between 1 − ϵ and 1 + ϵ times the number of satisfying truth assignments with probability at least 1 − δ. The running time of the algorithm is linear in the length of the formula times 1ϵ2 times ln(1δ). On the other hand, the problem of computing the exact answer for the DNF counting problem is known to be #P-complete, which implies that there is no polynomial time algorithm for the exact solution if P ≠ NP. This paper improves and gives new applications of some of the work previously reported. Variants of an ϵ, δ approximation algorithm for the DNF counting problem have been highly tailored to be especially efficient for the network reliability problems to which they are applied. In this paper the emphasis is on the development and analysis of a much more efficient ϵ, δ approximation algorithm for the DNF counting problem. The running time of the algorithm presented here substantially improves the running time of versions of this algorithm given previously. We give a new application of the algorithm to a problem which is relevant to physical chemistry and statistical physics. The resulting ϵ, δ approximation algorithm is substantially faster than the fastest known deterministic solution for the problem.

Markov Chain Algorithms for Planar Lattice Structures

Consider the following Markov chain, whose states are all domino tilings of a 2n× 2n chessboard: starting from some arbitrary tiling, pick a 2×2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes

A Monte-Carlo algorithm for estimating the permanent

90^{\rm o}

Approximating the permanent of graphs with large factors

in place. Repeat many times. This process is used in practice to generate a random tiling and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various two-dimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution.

Monte-Carlo algorithms for the planar multiterminal network reliability problem

Let A be an

Polytopes, permanents and graphs with large factors

n \times n

Markov chain algorithms for planar lattice structures

matrix with 0-1 valued entries, and let

Approximately counting up to four

{\operatorname{per}}(A)

Some theoretical results on the optimal pla folding problem

be the permanent of A. This paper describes a Monte-Carlo algorithm that produces a “good in the relative sense” estimate of