Makespan minimization on unrelated parallel machines with simple job-intersection structure and bounded job assignments

Abstract

Let there be a set J of n jobs and a set M of m parallel machines, where each job Jj takes pi,j∈Z+ time units on machine Mi and assume pi,j=∞ implies job Jj cannot be scheduled on machine Mi. In makespan minimization on unrelated parallel machines (R||Cmax), the goal is to schedule each job non-preemptively on a machine so as to minimize the makespan. A job-intersection graph GJ=(J,EJ) is an unweighted undirected graph where there is an edge {Jj,Jj′}∈EJ if there is a machine Mi such that both pi,j≠∞ and pi,j′≠∞. In this paper we consider two variants of R||Cmax where there are a small number of eligible jobs per machine. First, we prove that there are no approximation algorithms with approximation ratios less than 3/2 for R||Cmax when restricted to instances with job-intersection graphs belonging to some graph classes such as diamondless graphs and planar graphs, unless P=NP. Second, we match this lower bound by presenting a 3/2-approximation algorithm for R||Cmax restricted to instances with diamondless job-intersection graphs, and furthermore show that when GJ is triangle free R||Cmax is solvable in polynomial time. For R||Cmax restricted to instances when every machine can process at most ℓ jobs, we give an approximation algorithm with approximation ratios 3/2 and 5/3 for ℓ=3 and ℓ=4 respectively, a polynomial-time algorithm when ℓ=2, and prove that it is NP-hard to approximate the optimum solution within a factor less than 3/2 when ℓ≥3. In the special case where every pi,j∈{pj,∞}, called the restricted assignment problem, and there are only two job lengths pj∈{α,β} we present a (2−1/(ℓ−1))-approximation algorithm when ℓ≥3. In addition, we give a 2-approximation algorithm for the so-called unrelated graph balancing problem for ℓ=3 in the case where J is partitioned into b sets called bags, and no two jobs from the same bag can be scheduled on the same machine.