Document

**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

In this paper, we consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on m machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness of the instance, which is the maximum number of chains whose intervals between release date and deadline overlap.
We show that this scheduling problem with exact delays in unary is W[t]-hard for all t, when parameterized by the thickness, even when we have a single machine (m = 1). When parameterized by the number of chains, this problem is W[1]-complete when we have a single or a constant number of machines, and W[2]-complete when the number of machines is a variable. The problem with minimum delays, given in unary, parameterized by the number of chains (and as a simple corollary, also when parameterized by the thickness) is W[1]-hard for a single or a constant number of machines, and W[2]-hard when the number of machines is variable.
With a dynamic programming algorithm, one can show membership in XP for exact and minimum delays in unary, for any number of machines, when parameterized by thickness or number of chains. For a single machine, with exact delays in binary, parameterized by the number of chains, membership in XP can be shown with branching and solving a system of difference constraints. For all other cases for delays in binary, membership in XP is open.

Hans L. Bodlaender and Marieke van der Wegen. Parameterized Complexity of Scheduling Chains of Jobs with Delays. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{bodlaender_et_al:LIPIcs.IPEC.2020.4, author = {Bodlaender, Hans L. and van der Wegen, Marieke}, title = {{Parameterized Complexity of Scheduling Chains of Jobs with Delays}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {4:1--4:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.4}, URN = {urn:nbn:de:0030-drops-133075}, doi = {10.4230/LIPIcs.IPEC.2020.4}, annote = {Keywords: Scheduling, parameterized complexity} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Given a vertex-colored graph, we say a path is a rainbow vertex path if all its internal vertices have distinct colors. The graph is rainbow vertex-connected if there is a rainbow vertex path between every pair of its vertices. In the Rainbow Vertex Coloring (RVC) problem we want to decide whether the vertices of a given graph can be colored with at most k colors so that the graph becomes rainbow vertex-connected. This problem is known to be NP-complete even in very restricted scenarios, and very few efficient algorithms are known for it. In this work, we give polynomial-time algorithms for RVC on permutation graphs, powers of trees and split strongly chordal graphs. The algorithm for the latter class also works for the strong variant of the problem, where the rainbow vertex paths between each vertex pair must be shortest paths. We complement the polynomial-time solvability results for split strongly chordal graphs by showing that, for any fixed p ≥ 3 both variants of the problem become NP-complete when restricted to split (S₃,…,S_p)-free graphs, where S_q denotes the q-sun graph.

Paloma T. Lima, Erik Jan van Leeuwen, and Marieke van der Wegen. Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{lima_et_al:LIPIcs.MFCS.2020.63, author = {Lima, Paloma T. and van Leeuwen, Erik Jan and van der Wegen, Marieke}, title = {{Algorithms for the Rainbow Vertex Coloring Problem on Graph Classes}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {63:1--63:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.63}, URN = {urn:nbn:de:0030-drops-127331}, doi = {10.4230/LIPIcs.MFCS.2020.63}, annote = {Keywords: rainbow vertex coloring, permutation graphs, powers of trees} }