TY - JOUR

T1 - Exact and fixed parameter tractable algorithms for max-conflict-free coloring in hypergraphs∗

AU - Ashok, Pradeesha

AU - Dudeja, Aditi

AU - Kolay, Sudeshna

AU - Saurabh, Saket

N1 - Funding Information:
∗Received by the editors December 9, 2016; accepted for publication (in revised form) February 12, 2018; published electronically June 5, 2018. A preliminary version of this paper appeared in Algorithms and Computation, Lecture Notes in Comput. Sci. 9472, Springer, Berlin, 2015, pp. 271– 282. http://www.siam.org/journals/sidma/32-2/M110746.html Funding: The research leading to these results has received partial funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 306992. †International Institute of Information Technology, Bangalore, India (pradeesha@iiitb.ac.in). ‡Rutgers University, New Brunswick, NJ 08901 (ad1222@rutgers.edu). §Eindhoven University of Technology, Eindhoven, Netherlands (s.kolay@tue.nl). ¶The Institute of Mathematical Sciences, Taramani, India and Department of Informatics, University of Bergen, Bergen, Norway (saket@imsc.res.in).
Funding Information:
The research leading to these results has received partial funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 306992.
Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph H = (U, F), a conflict-free coloring of H refers to a vertex coloring where every hyperedge has a vertex with a unique color, distinct from all other vertices in the hyperedge. In this paper, we initiate a study of a natural maximization version of this problem, namely, Max-CFC: For a given hypergraph H and a fixed r ≥ 2, color the vertices of U using r colors so that the number of hyperedges that are conflict-free colored is maximized. By previously known hardness results for conflict-free coloring, this maximization version is NP-hard. We study this problem in the context of both exact and parameterized algorithms. In the parameterized setting, we study this problem with respect to a natural parameter—the solution size. In particular, the question we study is the following: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being the solution size k. We show that this problem is fixed parameter tractable by designing an algorithm with running time 2O(k log log k+k log r)(n + m)O (1) using a novel connection to the Unique Coverage problem and applying the method of color coding in a nontrivial manner. For the special case for hypergraphs induced by graph neighborhoods we give a polynomial kernel. Finally, we give an exact algorithm for Max-CFC running in O(2n+m) time. All our algorithms, with minor modifications, work for a stronger version of conflict-free coloring, Unique Maximum Coloring.

AB - Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph H = (U, F), a conflict-free coloring of H refers to a vertex coloring where every hyperedge has a vertex with a unique color, distinct from all other vertices in the hyperedge. In this paper, we initiate a study of a natural maximization version of this problem, namely, Max-CFC: For a given hypergraph H and a fixed r ≥ 2, color the vertices of U using r colors so that the number of hyperedges that are conflict-free colored is maximized. By previously known hardness results for conflict-free coloring, this maximization version is NP-hard. We study this problem in the context of both exact and parameterized algorithms. In the parameterized setting, we study this problem with respect to a natural parameter—the solution size. In particular, the question we study is the following: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being the solution size k. We show that this problem is fixed parameter tractable by designing an algorithm with running time 2O(k log log k+k log r)(n + m)O (1) using a novel connection to the Unique Coverage problem and applying the method of color coding in a nontrivial manner. For the special case for hypergraphs induced by graph neighborhoods we give a polynomial kernel. Finally, we give an exact algorithm for Max-CFC running in O(2n+m) time. All our algorithms, with minor modifications, work for a stronger version of conflict-free coloring, Unique Maximum Coloring.

KW - Conflict-free coloring

KW - FPT algorithms

KW - Maximization algorithms

KW - Unique-maximum coloring

UR - http://www.scopus.com/inward/record.url?scp=85049597373&partnerID=8YFLogxK

U2 - 10.1137/16M1107462

DO - 10.1137/16M1107462

M3 - Article

AN - SCOPUS:85049597373

SN - 0895-4801

VL - 32

SP - 1189

EP - 1208

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -