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Finding $b$-colorings Using Feedback Edges

Jakub Balabán

TL;DR

<3-5 sentence high-level summary> The paper investigates the b-coloring problem through the lens of fixed-parameter tractability (FPT) under structural parameters that capture proximity to tree-like graphs. It provides an FPT algorithm parameterized by the feedback edge number and, separately, by distance to co-cluster, by adapting and extending tree-based techniques and introducing novel constructs such as pivots, color plans, and signatures. A key contribution is a detailed, multi-layered framework that converts the global coloring problem into localized, analyzable subproblems on a small fen-core or a co-cluster modulator, then stitches these pieces back together to a valid b-coloring; this yields a 2^{O(p^2)}-time algorithm for fen and a 2^{2^{O(p)}}-time algorithm for distance-to-co-cluster. The results advance the parameterized complexity landscape for b-coloring and open paths toward understanding its behavior under other graph width and distance measures, while clarifying the limitations implied by tree-depth hardness.

Abstract

A $b$-coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The $b$-coloring problem, in which the task is to decide whether a graph admits a $b$-coloring with $k$ colors, is NP-complete in general but polytime solvable on trees. Moreover, it is known that $b$-coloring is in XP but W[$t$]-hard for all $t \in \mathbb{N}$ when parameterized by tree-width. In fact, only very few parameters, such as the vertex cover number, were known to admit an FPT algorithm for $b$-coloring. In this paper, we consider a more restrictive parameter measuring similarity to trees than tree-width, namely the feedback edge number, and show that $b$-coloring is fixed-parameter tractable under this parameterization. Our algorithm combines standard techniques used in parameterized algorithmics with the problem-specific ideas used in the polytime algorithm for trees. In addition, we present an FPT algorithm for $b$-coloring parameterized by distance to co-cluster, which is a parameter measuring similarity to complete multipartite graphs. Finally, we make several observations based on known results, including that $b$-coloring is W[$1$]-hard when parameterized by tree-depth.

Finding $b$-colorings Using Feedback Edges

TL;DR

<3-5 sentence high-level summary> The paper investigates the b-coloring problem through the lens of fixed-parameter tractability (FPT) under structural parameters that capture proximity to tree-like graphs. It provides an FPT algorithm parameterized by the feedback edge number and, separately, by distance to co-cluster, by adapting and extending tree-based techniques and introducing novel constructs such as pivots, color plans, and signatures. A key contribution is a detailed, multi-layered framework that converts the global coloring problem into localized, analyzable subproblems on a small fen-core or a co-cluster modulator, then stitches these pieces back together to a valid b-coloring; this yields a 2^{O(p^2)}-time algorithm for fen and a 2^{2^{O(p)}}-time algorithm for distance-to-co-cluster. The results advance the parameterized complexity landscape for b-coloring and open paths toward understanding its behavior under other graph width and distance measures, while clarifying the limitations implied by tree-depth hardness.

Abstract

A -coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The -coloring problem, in which the task is to decide whether a graph admits a -coloring with colors, is NP-complete in general but polytime solvable on trees. Moreover, it is known that -coloring is in XP but W[]-hard for all when parameterized by tree-width. In fact, only very few parameters, such as the vertex cover number, were known to admit an FPT algorithm for -coloring. In this paper, we consider a more restrictive parameter measuring similarity to trees than tree-width, namely the feedback edge number, and show that -coloring is fixed-parameter tractable under this parameterization. Our algorithm combines standard techniques used in parameterized algorithmics with the problem-specific ideas used in the polytime algorithm for trees. In addition, we present an FPT algorithm for -coloring parameterized by distance to co-cluster, which is a parameter measuring similarity to complete multipartite graphs. Finally, we make several observations based on known results, including that -coloring is W[]-hard when parameterized by tree-depth.

Paper Structure

This paper contains 29 sections, 49 theorems, 1 equation, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distance to Co-Cluster
  4. Initial setup
  5. Any minimal $\sigma$-coloring can be used
  6. Handling colors not used on $S$
  7. Finishing the proof
  8. Feedback Edge Number
  9. Initial setup
  10. General definition of pivots
  11. $S$-profiles
  12. Color plans and realizations
  13. Vertices in $B_\rho$ are candidates
  14. Failing $S$-profiles describe no $b$-colorings
  15. Computing a damage-free color realization
  16. ...and 14 more sections

Key Result

Theorem 1

Given an $n$-vertex graph $G$ with distance to cluster $p$, a minimum cluster-modulator $S\subseteq V(G)$ can be computed in time $\mathcal{O}(1.92^p \cdot n^2)$.

Figures (6)

  • Figure 1: Parameterized complexity of $b$-coloring under various structural parameters. A directed path from a parameter $\alpha$ to a parameter $\beta$ indicates that $\beta \le f(\alpha)$ for some computable function $f$. Green stands for FPT, blue is W[1]-hard and XP, red is paraNP-complete, and white means that it is unknown whether the problem is FPT or W[1]-hard under given parameterization. Our new results are marked with thick boundary, and the two parameters for which our FPT algorithms are non-trivial are written in red.
  • Figure 2: Outline of the algorithm for $b$-coloring parameterized by the feedback edge number. Each step refers to the relevant part of the proof.
  • Figure 3: An illustration of Definition \ref{['def:type']} with $S = \{u, v\}$ and $p = 2$. The edges in $G - S$ are drawn in gray. The left-hand set and the middle set are of the same type, namely $\{(\emptyset, 0), (\{u\}, 3), (\{v\}, 0), (\{u, v\}, 0)\}$.
  • Figure 5: A depiction of the fen-core $S$, see Definition \ref{['def:fen-core']}. If $k \le 9$, then the green vertices may be used as $b$-vertices. Note that in contrast to this figure, $G[S]$ may be disconnected.
  • Figure 7: Three possible cases, in which there is a $\rho$-pivot $u$ and a $\rho'$-pivot $u'$. The gray vertices are in $S_\rho$ and the square vertices are in $S$. The shift leading to $\rho'$ is depicted by the red arrow. Left: Both $u$ and $u'$ are $\chi$-tight, and for each $c \in [b+1, k]$, we have $\rho(c) \in N(u) \cup N(u')$. Middle and right: The vertex $w$ is $\chi$-tight and for each color $c' \in [b+1, k] \setminus \{c, d\}$, we have $\rho(c') \in N[w]$. It does not matter that $v$ is far from $u'$ because $\rho(d) \in N(u')$.
  • ...and 1 more figures

Theorems & Definitions (61)

  • Theorem 1: boral2016fast
  • Lemma 2
  • Lemma 3
  • Proposition 4: JaffkeLL24Davi_Andrade
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Definition 9
  • Definition 10
  • ...and 51 more