On the Relationship Between Combinatorial and LP-Based Approaches to NP-Hard Scheduling Problems

Enumerative approaches, such as branch-and-bound, to solv- ing 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.

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