On the relationship between combinatorial and LP-based lower bounds for NP-hard scheduling problems

Authors:
R. N. Uma;Joel Wein;David P. Williamson
Affiliations:
Department of Mathematics and Computer Science, North Carolina Central University, Durham, NC and Polytechnic University, Brooklyn, NY;Department of Computer Science, Polytechnic University, Brooklyn, NY and IBM T.J. Watson Research Center;School of Operations Research & Industrial Engineering and Computing & Information Science, Cornell University, Ithaca, NY
Venue:
Theoretical Computer Science - Approximation and online algorithms
Year:
2006

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Abstract

Enumerative approaches to solving optimization problems, such as branch and bound, require a subroutine that produces a lower bound on the value of the optimal solution. In the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear-programming (LP) relaxation of the problem or the solution to a combinatorial relaxation. In this paper we investigate, from a theoretical perspective, the relationship between several LP-based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled.We establish a number of facts about the relationship between these different sorts of lower bounds, including the equivalence of certain LP-based lower bounds for these problems to combinatorial lower bounds used in successful branch-and-bound algorithms. As a result, we obtain the first worst-case analysis of the quality of the lower bounds delivered by these combinatorial relaxations.