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DOI: 10.4230/LIPIcs.FSTTCS.2024.31
Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem

Authors: Rogers Mathew, Fahad Panolan, and Seshikanth

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
An edge q-coloring of a graph G is a coloring of its edges such that every vertex sees at most q colors on the edges incident on it. The size of an edge q-coloring is the total number of colors used in the coloring. Given a graph G and a positive integer t, the Maximum Edge q-Coloring problem is about whether G has an edge q-coloring of size t. Studies on this coloring problem were motivated by its application in the channel assignment problem in wireless networks. Goyal, Kamat, and Misra (MFCS 2013) studied Maximum Edge 2-Coloring from the perspective of parameterized complexity. Given a graph on n vertices, they considered the standard parameter t, the number of colors in an optimal edge 2-coloring, and the dual parameter 𝓁, where n-𝓁 is the number of colors in an optimal edge 2-coloring. They designed FPT algorithms for Maximum Edge 2-Coloring parameterized by t and 𝓁. In this paper, we revisit and study Maximum Edge 2-Coloring from the perspective of parameterized complexity and show the following results. 1) Let γ(G) denote the maximum matching size in a given graph G. It is easy to see that a maximum edge 2-coloring of G is of size at least γ(G). Goyal, Kamat, and Misra (MFCS 2013) had asked if there exists an FPT algorithm for Maximum Edge 2-Coloring parameterized by k, where k: = (size of a maximum edge 2-coloring of G) - γ(G). We show that Maximum Edge 2-Coloring parameterized by k is W[1] hard. 2) On the positive side, we show that there is an algorithm that, given a graph G on n vertices and a tree decomposition of width tw, runs in time 2^{O(qtw log {q tw})}n and outputs a maximum edge q-coloring of G.
Cite as

Rogers Mathew, Fahad Panolan, and Seshikanth. Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mathew_et_al:LIPIcs.FSTTCS.2024.31,
  author =	{Mathew, Rogers and Panolan, Fahad and Seshikanth},
  title =	{{Parameterized Algorithms and Hardness for the Maximum Edge q-Coloring Problem}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{31:1--31:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.31},
  URN =		{urn:nbn:de:0030-drops-222202},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.31},
  annote =	{Keywords: FPT algorithm, Edge coloring, Treewidth, W\lbrack1\rbrack-hardness}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.19
Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs

Authors: Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, and Rogers Mathew

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
A Conflict-Free Open Neighborhood coloring, abbreviated CFON^* coloring, of a graph G = (V,E) using k colors is an assignment of colors from a set of k colors to a subset of vertices of V(G) such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^* coloring using k colors is called the CFON^* chromatic number of G, denoted by χ_{ON}^*(G). The analogous notion for closed neighborhood is called CFCN^* coloring and the analogous parameter is denoted by χ_{CN}^*(G). The problem of deciding whether a given graph admits a CFON^* (or CFCN^*) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper. - For k ≥ 3, we show that if G is a K_{1,k}-free graph then χ_{ON}^*(G) = O(k²log Δ), where Δ denotes the maximum degree of G. Dębski and Przybyło in [J. Graph Theory, 2021] had shown that if G is a line graph, then χ_{CN}^*(G) = O(log Δ). As an open question, they had asked if their result could be extended to claw-free (K_{1,3}-free) graphs, which are a superclass of line graphs. Since it is known that the CFCN^* chromatic number of a graph is at most twice its CFON^* chromatic number, our result positively answers the open question posed by Dębski and Przybyło. - We show that if the minimum degree of any vertex in G is Ω(Δ/{log^ε Δ}) for some ε ≥ 0, then χ_{ON}^*(G) = O(log^{1+ε}Δ). This is a generalization of the result given by Dębski and Przybyło in the same paper where they showed that if the minimum degree of any vertex in G is Ω(Δ), then χ_{ON}^*(G)= O(logΔ). - We give a polynomial time algorithm to compute χ_{ON}^*(G) for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether the CFON^* chromatic number can be computed in polynomial time on interval graphs. - We explore biconvex graphs, a subclass of bipartite graphs and give a polynomial time algorithm to compute their CFON^* chromatic number. This is interesting as Abel et al. [SIDMA, 2018] had shown that it is NP-complete to decide whether a planar bipartite graph G has χ_{ON}^*(G) = k where k ∈ {1, 2, 3}.
Cite as

Sriram Bhyravarapu, Subrahmanyam Kalyanasundaram, and Rogers Mathew. Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhyravarapu_et_al:LIPIcs.MFCS.2022.19,
  author =	{Bhyravarapu, Sriram and Kalyanasundaram, Subrahmanyam and Mathew, Rogers},
  title =	{{Conflict-Free Coloring on Claw-Free Graphs and Interval Graphs}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.19},
  URN =		{urn:nbn:de:0030-drops-168173},
  doi =		{10.4230/LIPIcs.MFCS.2022.19},
  annote =	{Keywords: Conflict-free coloring, Interval graphs, Bipartite graphs, Claw-free graphs}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.176
Cubicity, Degeneracy, and Crossing Number

Authors: Abhijin Adiga, L. Sunil Chandran, and Rogers Mathew

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
A k-box B=(R_1,R_2,...,R_k), where each R_i is a closed interval on the real line, is defined to be the Cartesian product R_1 X R_2 X ... X R_k. If each R_i is a unit length interval, we call B a k-cube. Boxicity of a graph G, denoted as box(G), is the minimum integer k such that G is an intersection graph of k-boxes. Similarly, the cubicity of G, denoted as cub(G), is the minimum integer k such that G is an intersection graph of k-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan. Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/0607092, 2006.] that, for a graph G with maximum degree \Delta, cub(G) <= \lceil 4(\Delta +1) ln n\rceil. In this paper we show that, for a k-degenerate graph G, cub(G) <= (k+2) \lceil 2e log n \rceil. Since k is at most \Delta and can be much lower, this clearly is a stronger result. We also give an efficient deterministic algorithm that runs in O(n^2k) time to output a 8k(\lceil 2.42 log n\rceil + 1) dimensional cube representation for G. The crossing number of a graph G, denoted as CR(G), is the minimum number of crossing pairs of edges, over all drawings of G in the plane. An important consequence of the above result is that if the crossing number of a graph G is t, then box(G) is O(t^{1/4}{\lceil log t\rceil}^{3/4}) . This bound is tight upto a factor of O((log t)^{3/4}). Let (P,\leq) be a partially ordered set and let G_{P} denote its underlying comparability graph. Let dim(P) denote the poset dimension of P. Another interesting consequence of our result is to show that dim(P) \leq 2(k+2) \lceil 2e \log n \rceil, where k denotes the degeneracy of G_{P}. Also, we get a deterministic algorithm that runs in O(n^2k) time to construct a 16k(\lceil 2.42 log n\rceil + 1) sized realizer for P. As far as we know, though very good upper bounds exist for poset dimension in terms of maximum degree of its underlying comparability graph, no upper bounds in terms of the degeneracy of the underlying comparability graph is seen in the literature.
Cite as

Abhijin Adiga, L. Sunil Chandran, and Rogers Mathew. Cubicity, Degeneracy, and Crossing Number. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 176-190, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{adiga_et_al:LIPIcs.FSTTCS.2011.176,
  author =	{Adiga, Abhijin and Chandran, L. Sunil and Mathew, Rogers},
  title =	{{Cubicity, Degeneracy, and Crossing Number}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{176--190},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.176},
  URN =		{urn:nbn:de:0030-drops-33428},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.176},
  annote =	{Keywords: Degeneracy, Cubicity, Boxicity, Crossing Number, Interval Graph, Intersection Graph, Poset Dimension, Comparability Graph}
}
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