eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-18
6:1
6:14
10.4230/LIPIcs.MFCS.2020.6
article
Complexity of Computing the Anti-Ramsey Numbers for Paths
Akhoondian Amiri, Saeed
1
https://orcid.org/0000-0002-7402-2662
Popa, Alexandru
2
3
https://orcid.org/0000-0003-3364-1210
Roghani, Mohammad
4
5
https://orcid.org/0000-0001-8247-3773
Shahkarami, Golnoosh
6
https://orcid.org/0000-0002-6169-7337
Soltani, Reza
4
5
https://orcid.org/0000-0002-8875-5023
Vahidi, Hossein
6
https://orcid.org/0000-0002-0040-1213
University of Cologne, Germany
University of Bucharest, Romania
National Institute of Research and Development in Informatics, Bucharest, Romania
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Sharif University of Technology, Teheran, Iran
MPI for Informatics, Saarland Informatics Campus, Graduate School of Computer Science, Saarbrücken, Germany
The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color.
Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant.
We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k.
Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol170-mfcs2020/LIPIcs.MFCS.2020.6/LIPIcs.MFCS.2020.6.pdf
Coloring
Anti-Ramsey
Approximation
NP-hard
Algorithm
ETH