On the relationship between combinatorial and LP-based approaches to NP-hard scheduling problems

Enumerative approaches, such as branch-and-bound, to solving optimization problems 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 relaxation of the problem or the solution of a combinatorial relaxation. In this paper we investigate, from both a theoretical and practical perspective, the relationship between several linear-programming based lower bounds and combinatorial lower bounds for two 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 linear-programming-based lower bounds for both of 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 bound delivered by these combinatorial relaxations, We then give an empirical evaluation of the strength of the various lower bounds and heuristics. This extends and puts in a broader context a recent experimental evaluation by Savelsbergh and the authors of the empirical strength of bath heuristics and lower bounds based on different LP-relaxations of a single-machine scheduling problem. We observe that on most kinds of synthetic data used in experimental studies a simple heuristic, used in successful combinatorial branch-and-bound algorithms far the problem, outperforms on average all of the LP-based heuristics. However, we identify other classes of problems on which the LP-based heuristics are superior, and report on experiments that give a qualitative sense of the range of dominance of each. Finally, we consider the impact of local improvement on the solutions.