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Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Abstract

In the paper we consider the problem of scheduling nn identical jobs on 4 uniform machines with speeds s1≥s2≥s3≥s4,s_1 \geq s_2 \geq s_3 \geq s_4, respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree Δ\Delta, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if s1=s2=s3s_1=s_2=s_3. If, however, Δ≤4\Delta \leq 4 and s1≥12s2s_1 \geq 12 s_2, s2=s3=s4s_2=s_3=s_4, then the problem can be solved to optimality in time O(n1.5)O(n^{1.5}). The same algorithm returns a solution of value at most 2 times optimal provided that s1≥2s2s_1 \geq 2s_2. Finally, we study the case s1≥s2≥s3=s4s_1 \geq s_2 \geq s_3=s_4 and give an O(n1.5)O(n^{1.5})-time 32/1532/15-approximation algorithm in all such situations

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