Total completion time minimization for scheduling with incompatibility cliques

作     者：Jansen, Klaus Lassota, Alexandra Maack, Marten Pikies, Tytus 

作者机构：Department of Computer Science Faculty Of Engineering Kiel University Germany Department of Algorithms and System Modelling ETI Faculty Gdańsk University of Technology Poland 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2020年

核心收录：

主　　题：Polynomial approximation 

摘      要：This paper considers parallel machine scheduling with incompatibilities between jobs. The jobs form a graph and no two jobs connected by an edge are allowed to be assigned to the same machine. In particular, we study the case where the graph is a collection of disjoint cliques. Scheduling with incompatibilities between jobs represents a well-established line of research in scheduling theory and the case of disjoint cliques has received increasing attention in recent years. While the research up to this point has been focused on the makespan objective, we broaden the scope and study the classical total completion time criterion. In the setting without incompatibilities, this objective is well known to admit polynomial time algorithms even for unrelated machines via matching techniques. We show that the introduction of incompatibility cliques results in a richer, more interesting picture. Scheduling on identical machines remains solvable in polynomial time, while scheduling on unrelated machines becomes APX-hard. We identify several more subcases of theses problems that are polynomial time solvable or NP-hard, respectively. Furthermore, we study the problem under the paradigm of fixed-parameter tractable algorithms (FPT). In particular, we consider a problem variant with assignment restrictions for the cliques rather than the jobs. We prove that it is NP-hard and can be solved in FPT time with respect to the number of cliques. Moreover, we show that the problem on unrelated machines can be solved in FPT time for reasonable parameters, e.g., the parameter pair: number of machines and maximum processing time. The latter result is a natural extension of known results for the case without incompatibilities and can even be extended to the case of total weighted completion time. All of the FPT results make use of n-fold Integer Programs that recently have received great attention by proving their usefulness for scheduling problems. Copyright © 2020, The Authors. All rights