Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

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,...

show abstract

“…However, to the best of our knowledge little work has been done on scheduling problems with uniform machines involved (cf. [6,7]). …”

mentioning

confidence: 99%

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

Furmańczyk

Kubale

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

View full text Add to dashboard Cite

show abstract

“…In what follows, such a color class V 1 will be called non-equitable. These two models of graph coloring are motivated by applications in multiprocessor scheduling of unit-execution time jobs [7,8].…”

Section: Introductionmentioning

confidence: 99%

Tight bounds on the complexity of semi-equitable coloring of cubic and subcubic graphs

Furmańczyk

Kubale

2018

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

A k-coloring of a graph G = (V, E) is called semi-equitable if there exists a partition of its vertex set into independent subsets V 1 , . . . , V k in such a way thatThe color class V 1 is called non-equitable. In this note we consider the complexity of semi-equitable k-coloring, k ≥ 4, of the vertices of a cubic or subcubic graph G. In particular, we show that, given a n-vertex subcubic graph G and constants > 0, k ≥ 4, it is NP-complete to obtain a semi-equitable k-coloring of G whose non-equitable color class is of size s if s ≥ n/3 + n, and it is polynomially solvable if s ≤ n/3. *

show abstract

“…Furmańczyk [4] mentions a specific application of this type of scheduling problem, namely, assigning university courses to time slots in a way that avoids scheduling incompatible courses at the same time and spreads the courses evenly among the available time slots. Also, the application of equitable coloring in scheduling of jobs on uniform machines was considered in [7] and [8].…”

Section: Introductionmentioning

confidence: 99%

Equitable colorings of corona multiproducts of graphs

Furmańczyk¹,

Kubale²,

Mkrtchyan³

2017

Discuss. Math. Graph Theory

Self Cite

View full text Add to dashboard Cite

A graph G is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest integer k for which such a coloring exists is known as the equitable chromatic number of G and it is denoted by χ = (G).In this paper the problem of determinig the value of equitable chromatic number for multicoronas of cubic graphs G • l H is studied. The problem of ordinary coloring of multicoronas of cubic graphs is solvable in polynomial time. The complexity of equitable coloring problem is an open question for these graphs. We provide some polynomially solvable cases of cubical multicoronas and give simple linear time algorithms for equitable coloring of such graphs which use at most χ = (G • l H) + 1 colors in the remaining cases.

show abstract

Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

Cited by 17 publications

References 17 publications

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

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

Tight bounds on the complexity of semi-equitable coloring of cubic and subcubic graphs

Equitable colorings of corona multiproducts of graphs

Contact Info

Product

Resources

About