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 j takes \(p_{i,j} \in \mathbb {Z}^+\) time units on machine i and assume \(p_{i,j}=\infty \) implies job j cannot be scheduled on machine i. In makespan minimization on unrelated parallel machines (\(R||C_{max}\)), the goal is to schedule each job non-preemptively on a machine so as to minimize the makespan. A job-intersection graph \(G_J=(J,E_J)\) is an unweighted undirected graph where there is an edge \(\{j,j'\} \in E_J\) if there is a machine i such that both \(p_{i,j}\ne \infty \) and \(p_{i,j'} \ne \infty \). In this paper we consider two variants of \(R||C_{max}\) where there are a small number of eligible jobs per machine. First, we prove that there is no approximation algorithm with approximation ratio better than 3/2 for \(R||C_{max}\) when restricted to instances where the job-intersection graph contains no diamonds, unless Open image in new window . Second, we match this lower bound by presenting a 3/2-approximation algorithm for this special case of \(R||C_{max}\), and furthermore show that when \(G_J\) is triangle free \(R||C_{max}\) is solvable in polynomial time. For \(R||C_{max}\) restricted to instances when every machine can process at most \(\ell \) jobs, we give approximation algorithms with approximation ratios 3/2 and 5/3 for \(\ell =3\) and \(\ell =4\) respectively, a polynomial-time algorithm when \(\ell =2\), and prove that it is Open image in new window -hard to approximate the optimum solution within a factor less than 3/2 when \(\ell \ge 3\). In the special case where every \(p_{i,j} \in \{p_j, \infty \}\), called the restricted assignment problem, and there are only two job lengths \(p_j \in \{\alpha ,\beta \}\) we present a \((2-1/(\ell -1))\)-approximation algorithm when \(\ell \ge 3\).