Implementation of Approximation Algorithms for the Max-Min Resource Sharing Problem

Abstract

We implement the algorithm for the max-min resource sharing problem described in [7], using a new line search technique for determining a suitable step length. Our line search technique uses a modified potential function that is less costly to evaluate, thus heuristically simplifying the computation. Observations concerning the quality of the dual solution and oscillating behavior of the algorithm are made. First numerical observations are briefly discussed. In particular we study a certain class of linear programs, namely the computational bottleneck of an algorithm from [8] for solving strip packing with an approach from [10, 13]. For these, we obtain practical running times. Our implementation is able to solve instances for small accuracy parameters ε for which the methods proposed in theory are out of practical interest. More precisely, the technique from improves the known runtime bound of O(M ⁶ln ²(Mn/(at))+M ⁵ n/t+ln (Mn/(at))) to the more favourable bound O(M(ε ^{− − 3}(ε ^{− − 2}+ln M)+M(ε ^{− − 2}+ln M))), where n denotes the number of items, M the number of distinct item widths, a the width of the narrowest item and t is a desired additive tolerance. Keywords: Algorithm Engineering, Implementation, Testing, Evaluation and Fine-tuning, Mathematical Programming.

Research of the authors was supported in part by EU Project CRESCCO, Critical Resource Sharing for Cooperation in Complex Systems, IST-2001-33135 and by DFG Project, Entwicklung und Analyse von Approximativen Algorithmen für Gemischte und Verallgemeinerte Packungs- und Überdeckungsprobleme, JA 612/10-1.