# Makespan Minimization on Unrelated Parallel Machines with Simple Job-Intersection Structure and Bounded Job Assignments

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## 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 . 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 -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 $$\ell \ge 3$$.

## Keywords

Makespan minimization Unrelated parallel machines Approximation algorithms Restricted assignment Bounded job assignments Job-intersection graphs

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## Authors and Affiliations

• Daniel R. Page
• 1
• Roberto Solis-Oba
• 1
• Marten Maack
• 2
1. 1.Department of Computer ScienceWestern UniversityLondonCanada
2. 2.Department of Computer ScienceUniversity of KielKielGermany