Scheduling with Communication Delays via LP Hierarchies and Clustering
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We consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by P|prec, c | C_max, if two dependent jobs are scheduled on different machines, then at least c units of time must pass between their executions. Despite its relevance to many applications, this model remains one of the most poorly understood in scheduling theory. Even for a special case where an unlimited number of machines is available, the best known approximation ratio is 2/3 · (c+1), whereas Graham's greedy list scheduling algorithm already gives a (c+1)-approximation in that setting. An outstanding open problem in the top-10 list by Schuurman and Woeginger and its recent update by Bansal asks whether there exists a constant-factor approximation algorithm. In this work we give a polynomial-time O(log c ·log m)-approximation algorithm for this problem, where m is the number of machines and c is the communication delay. Our approach is based on a Sherali-Adams lift of a linear programming relaxation and a randomized clustering of the semimetric space induced by this lift.
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