TY - GEN

T1 - Exact and FPT algorithms for max-conflict free coloring in hypergraphs

AU - Ashok, Pradeesha

AU - Dudeja, Aditi

AU - Kolay, Sudeshna

N1 - 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 no. 306992.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.

PY - 2015/1/1

Y1 - 2015/1/1

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 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 the natural parameter, the solution size. In particular, we study the following question: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being k. We show that this problem is FPT 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 non-trivial manner. For the special case for hypergraphs induced by graph neighbourhoods 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 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 the natural parameter, the solution size. In particular, we study the following question: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being k. We show that this problem is FPT 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 non-trivial manner. For the special case for hypergraphs induced by graph neighbourhoods 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.

UR - http://www.scopus.com/inward/record.url?scp=84952020244&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-48971-0_24

DO - 10.1007/978-3-662-48971-0_24

M3 - Conference contribution

AN - SCOPUS:84952020244

SN - 9783662489703

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 271

EP - 282

BT - Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings

A2 - Elbassioni, Khaled

A2 - Makino, Kazuhisa

PB - Springer Verlag

T2 - 26th International Symposium on Algorithms and Computation, ISAAC 2015

Y2 - 9 December 2015 through 11 December 2015

ER -