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

Abstract

Abstract. In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s 1 ¸ s 2 ¸ s 3 ¸ 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 ∆, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s 1 = s 2 = s 3 . If, however, ∆ • 4 and s 1 ¸ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5 ). The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ¸ 2s 2 . Finally, we study the case s 1 ¸ s 2 ¸ s 3 = s 4 and give a 32/15-approximation algorithm running also in O(n 1.5 ) time.Key words: equitable coloring, NP-hardness, polynomial algorithm, scheduling, uniform machine. edges correspond to pairs of jobs being in conflict, is a bipartite graph (without isolated vertices). For example, all graphs in our figures are bipartite. Notice that two jobs being in conflict may be executed in intersecting time intervals. A load of k jobs on M i machine requires the processing time k/s i , and all the jobs are ready for processing at the same time. Alternatively, if a load on M i is not given explicitly, we are using the notation C(M i ) to mean the schedule length on the machine M i . By definition, each load forms an independent set (color) in G. Therefore, we will be using the terms job/vertex and load/color/independent set interchangeably. Since all the tasks have to be executed, the problem is to find a 4-coloring, i.e. a decomposition of G into 4 independent sets I 1 , I 2 , I 3 , and I 4 such that the schedule length C max = maxfjI i j/s i : i = 1, …, 4g is minimized, in sym- Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machinesThere are several papers devoted to chromatic scheduling in the presence of mutual exclusion constraints. Boudhar in [1,2] studied the problem of batch scheduling with complements of bipartite and split graphs, respectively. Finke et al. [3] considered this problem with complements of interval graphs. Other models of batch scheduling with incompatibility constraints were studied in [4,5]. In all the papers the authors assumed identical parallel machines. However, to the best of our knowledge little work has been done on scheduling problems with uniform machines involved (cf. [6,7]).The rest of this paper is organized as follows. In Section 2 we show that the general problem is NP-hard even if s 1 = s 2 = s 3 . In Section 3 we show that if s 1 ¸ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5 ) provided that the degree of G is ∆ • 4. The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ¸ 2s 2 , s 2 = s 3 = s 4 . In Section 4 we study the case s 1 ¸ s 2 ¸ s 3 = s 4 and give an O(n 1.5 )-time 32/15-approximation algorithm in all such cases. Finally,...

Scheduling on uniform and unrelated machines with bipartite incompatibility graphs

“…Hence, we establish the complexity status of scheduling on uniform machines. This result justifies the research on families of graphs more restrictive than just being bipartite, like bipartite with bounded degree considered in [7], [8], and [23]. Moreover, due to our best…”

Mixed Graph Colorings: A Historical Review

“…As demonstrated in [84,85], colorings of edges in the graph may be used to model a certain job-shop scheduling system consisting of unit-time jobs assigned to specific pairs of machines. In the case of the mixed graph G = (V, A, E), it is convenient to look upon arc (v i , v j ) ∈ A as a unit-time data transmission from machine v i to machine v j requiring the cooperation of machines v i and v j , which cannot simultaneously process other jobs.…”

Scheduling of Identical Jobs with Bipartite Incompatibility Graphs on Uniform Machines. Computational Experiments

“…The authors of (Furmańczyk and Kubale, 2017a) also considered bisubquartic graphs and m = 4 machines. They proved that, if s 1 12s 2 and s 2 = s 3 = s 4 , then problem Q4|U ET, bisubquartic|C max can be solved to optimality in time O(n 1.5 ).…”

“…Therefore, we have the following: Theorem 3. (Furmańczyk and Kubale, 2017a) If s 1 12s 2 and s 2 = s 3 = s 4 then the problems Q4|U ET, bisubquartic|C max and Q4|U ET, bisubquartic| C i can be solved in O(n 1.5 ) time.…”