Markov chains, computer proofs, and average-case analysis of best fit bin packing

Abstract

Many complex Coffrnan, Jr. 1, D. S. Johnsonl, P. W. Shorl, R. R. Weber2 pmcessea can be modeled by (countably) infinite, multidmens~onst Markov chains. Unfortunately, cnrnmt theoretical techniques for analyzing infinite Markov chains are for the most part limited to three or fewer dimensions. In this paper we propose a computer-aided approach to the analysis of higher-dimensional domains, using several open problems about the average-case behavior of the Best Fit bin packing algorithm as case studies. We show how to use dynamic and liiear programming to construct potential functions thal when applied to suitably modified multi-step versions of our original Markov chain, yield drifts that are bounded away fmm O. This enables us to completely classify the expected behavior of Best Fit under discrete uniform distributions U{J, K) when K is small. (Under U{ J, K}, the allowed item sizes are i/K, 1 S i S J, with all J possibilities equally likely.) In addition, we can answer yes to the long-standing open question of whether there exist distributions of thii form for which Best Fit yields linearly-growing waste. The proof of the latter theorem relies on a 24-hour computation, and although its validity does not depend on the linear programmingpackage we used, it does tely on the correctness of our dynamic progr smming code and of our computer’s implementation of the IEEE floating point standard.