A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Abstract

We consider the following single-machine scheduling problem, which is often denoted 1|| f j : we are given n jobs to be scheduled on a single machine, where each job j has an integral processing time p j , and there is a nondecreasing, nonnegative cost function f j (C j ) that specifies the cost of finishing j at time C j ; the objective is to minimize n j=1 f j (C j ). Bansal & Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the LP is less than 4. Finally, we show how the technique can be adapted to yield, for any ǫ > 0, a (4+ǫ)-approximation algorithm for this problem.

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“…Relation with other work. Knapsack cover inequalities and their generalizations such as flow cover inequalities were used as a systematic way to strengthen LP formulations of other (seemingly unrelated) problems [15,14,33,4,5,16,6,20,34,19,22]. By strengthening we mean that one would start with a polynomial-size LP formulation with a potentially unbounded integrality gap for some problem of interest, and then show that adding (adaptations of) knapsack cover inequalities reduces this integrality gap.…”

“…In the special case of identical release time of the jobs, their LP formulation yields a 16-approximation algorithm. This constant-factor approximation was later improved by Cheung et al [19] to a (4 + ε)-approximation, where the authors added the knapsack cover inequalities directly to the LP formulation of the scheduling problem, i. e., without resorting to the intermediate geometric cover problem as in [6]. For both the GSP and its special case, our method yields an LP formulation whose size is quasi-polynomial in n, and polynomial in both log P and logW , where W is the maximum increase in the cost function of a job at any point in time.…”

Approximation algorithm with constant ratio for stochastic prize-collecting Steiner tree problem

“…The primal-dual scheme is a useful technique in designing approximation algorithms for NP-hard problems [9,13,21,29]. Goemans and Williamson [17] used the primal-dual scheme to derive a 2-approximation for the rooted PCST, where n = |V |.…”