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Fast FPT Algorithms for Grundy Number on Dense Graphs

Sina Ghasemi Nezhad, Maryam Moghaddas, Fahad Panolan

TL;DR

The paper studies the Grundy coloring problem, i.e., the Grundy chromatic number defined by the worst-case performance of the first-fit coloring, on dense graphs parameterized by cluster modulators. It develops an array of fixed-parameter tractable approaches: a clique-modulator based algorithm with running time $O(2^{O(r^2)}+n+m)$ and a kernel of size $O(r2^r)$; a max-flow based method for the 2-cluster modulator case achieving $O(2^{O(r^2)}n^6)$; and a general ILP-based framework for $k$-cluster modulators yielding time $O(2^{O(kr^2)} p^{2.5p+o(p)})$ with $p=O(2^{kr} r)$. These results combine structural insights on Grundy colorings with representative-set reductions, max-flow formulations, and IPT-amenable ILP techniques to obtain FPT algorithms for several dense-graph modulators. Overall, the work advances efficient exact algorithms for Grundy coloring in graph classes characterized by a small cluster modulator, with kernelization and ILP-based generalizations supporting broader applicability. The methods have potential impact on parameterized algorithms for adversarial coloring problems in dense networks and related graph classes.

Abstract

In this paper, we investigate the \textsc{Grundy Coloring} problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the first-fit coloring of a graph in the worst-case vertex ordering, is known to be $W[1]$-hard when parameterized by the number of colors required by the most adversarial ordering. We focus on fixed-parameter tractable (FPT) algorithms for solving this problem on graph classes characterized by dense substructures, specifically those with a cluster modulator. A cluster modulator is a vertex subset whose removal results in a cluster graph (a disjoint union of cliques). We present FPT algorithms for graphs where the cluster graph consists of one, two, or $k$ cliques, leveraging the cluster modulator's properties to achieve the best-known FPT runtimes, parameterized by both the modulator's size and the number of cliques.

Fast FPT Algorithms for Grundy Number on Dense Graphs

TL;DR

The paper studies the Grundy coloring problem, i.e., the Grundy chromatic number defined by the worst-case performance of the first-fit coloring, on dense graphs parameterized by cluster modulators. It develops an array of fixed-parameter tractable approaches: a clique-modulator based algorithm with running time and a kernel of size ; a max-flow based method for the 2-cluster modulator case achieving ; and a general ILP-based framework for -cluster modulators yielding time with . These results combine structural insights on Grundy colorings with representative-set reductions, max-flow formulations, and IPT-amenable ILP techniques to obtain FPT algorithms for several dense-graph modulators. Overall, the work advances efficient exact algorithms for Grundy coloring in graph classes characterized by a small cluster modulator, with kernelization and ILP-based generalizations supporting broader applicability. The methods have potential impact on parameterized algorithms for adversarial coloring problems in dense networks and related graph classes.

Abstract

In this paper, we investigate the \textsc{Grundy Coloring} problem for graphs with a cluster modulator, a structure commonly found in dense graphs. The Grundy chromatic number, representing the maximum number of colors needed for the first-fit coloring of a graph in the worst-case vertex ordering, is known to be -hard when parameterized by the number of colors required by the most adversarial ordering. We focus on fixed-parameter tractable (FPT) algorithms for solving this problem on graph classes characterized by dense substructures, specifically those with a cluster modulator. A cluster modulator is a vertex subset whose removal results in a cluster graph (a disjoint union of cliques). We present FPT algorithms for graphs where the cluster graph consists of one, two, or cliques, leveraging the cluster modulator's properties to achieve the best-known FPT runtimes, parameterized by both the modulator's size and the number of cliques.

Paper Structure

This paper contains 5 sections, 19 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grundy Coloring with a Clique Modulator
  4. Grundy Coloring with a 2-Cluster Modulator
  5. Solving the $k$-Cluster Modulator Case

Key Result

lemma thmcounterlemma

Let $G$ be a graph and $u,v\in V(G)$ such that $N_G[u] = N_{G}[v]$. Let $\sigma$ be a permutation of $V(G)$ and $\hat{\sigma}$ is the permutation obatined from $\sigma$ by swapping the vertices $u$ and $v$ in $\sigma$. Then, the first-fit colorings of $G$ with respect to $\sigma$ and $\hat{\sigma}$

Figures (1)

  • Figure 1: An overview of the flow network for extendable checking algorithm.

Theorems & Definitions (38)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 28 more