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

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In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s1 ≥ s2 ≥ s3 ≥ s4, 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 Δ, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s1 = s2 = s3. If, however, Δ ≤ 4 and s1 ≥ 12s2, s2 = s3 = s4, then the problem can be solved to optimality in time O(n1.5). The same algorithm returns a solution of value at most 2 times optimal provided that s1 ≥ 2s2. Finally, we study the case s1 ≥ s2 ≥ s3 = s4 and give a 32/15-approximation algorithm running also in O(n1.5) time.

[1] M. Boudhar, “Scheduling a batch processing machine with bipartite compatibility graphs”, Math. Methods Oper. Res. 57, 513-527 (2003).

[2] M. Boudhar, “Scheduling on a batch processing machine with split compatibility graphs”, J. Math. Modell. Algorithms 4, 391- 407 (2005).

[3] G. Finke, V. Jost, M. Queyranne, A. Sebó, “Batch processing with interval graph compatibilities between tasks”, Disc. Appl. Math. 156, 556-568 (2008).

[4] M. Demange, D. de Werra, J. Monnot, V.Th. Paschos, “Time slot scheduling of compatible jobs”, J. Scheduling 10, 111-127 (2007).

[5] D. de Werra, M. Demange, J. Monnot, V.Th. Paschos, “A hypocoloring model for batch scheduling”, Disc. Appl. Math. 146, 3-26 (2005).

[6] H. Furmańczyk, M. Kubale, “Scheduling of unit-length jobs with cubic Incompatibility graphs on three uniform machines”, Disc. Appl. Math., in print, available online 22 Feb. 2016.

[7] S.-S. Li, Y.-Z. Zhang, “Serial batch scheduling on uniform parallel machines to minimize total completion time”, Inf. Process. Lett. 114, 692-695 (2014).

[8] B.-L. Chen, C.-H. Yen, “Equitable Δ-coloring of graphs”, Disc. Math. 312, 1512-1517 (2012).

[9] J.E. Hopcroft, R.M. Karp, “An n5/2 algorithm for maximum matchings in bipartite graphs”, SIAM J. Comput. 2, 225-231 (1973).

[10] H.L. Bodlaender, K. Jansen, “On the complexity of scheduling incompatible jobs with unit-times”, LNCS 711, 291-300 (1993).

[11] D. König, “Gráfok és mátrixok”, Matematikai és Fizikai Lapok 38, 116-119 (1931), [in Hungarian].

[12] M. Kubale, “Interval vertex-coloring of a graph with forbidden colors”, Disc. Math. 74, 125-136 (1989).

Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

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IMPACT FACTOR 2016: 1.156
5-year IMPACT FACTOR: 1.238

CiteScore 2016: 1.50

SCImago Journal Rank (SJR) 2016: 0.457
Source Normalized Impact per Paper (SNIP) 2016: 1.239


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