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Residue Domination in Bounded-Treewidth Graphs

Jakob Greilhuber, Philipp Schepper, Philip Wellnitz

TL;DR

This work analyzes the (σ,ρ)-Dom-Set problem when σ and ρ are residue classes modulo m≥2, focusing on a treewidth parameterization. It develops a fixed-parameter algorithm running in time ${\mathrm m}^{\mathrm{tw}}\cdot n^{O(1)}$ that solves the decision, minimization, maximization, and exact variants, by a novel DP on tree decompositions that leverages sparse languages and weight-vector compression. The authors establish matching SETH-based lower bounds for pathwidth (and thus treewidth), showing that, for the difficult residue-class pairs, no $(m-\varepsilon)^{\mathrm{pw}}\cdot n^{O(1)}$-time algorithm exists. They also address special parity cases (Lights Out variants) and prove lower bounds for Reflexive-AllOff and AllOff, complementing the upper bounds. Overall, the paper provides a near-complete picture of the fixed-parameter tractability frontier for this broad class of generalized domination problems on bounded-treewidth graphs, and offers techniques (sparse languages, defining sets, and fast joins) that may apply to other non-finite set families.

Abstract

For the vertex selection problem $(σ,ρ)$-DomSet one is given two fixed sets $σ$ and $ρ$ of integers and the task is to decide whether we can select vertices of the input graph such that, for every selected vertex, the number of selected neighbors is in $σ$ and, for every unselected vertex, the number of selected neighbors is in $ρ$ [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when $σ$ and $ρ$ are two (potentially different) residue classes modulo $m\ge 2$. We study the problem parameterized by treewidth and present an algorithm that solves in time $m^{tw} \cdot n^{O(1)}$ the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case $m \ge 3$ not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no $(m-ε)^{pw} \cdot n^{O(1)}$-time algorithm parameterized by pathwidth $pw$, unless SETH fails. For $m = 2$, we extend these bounds to the minimization version as the decision version is efficiently solvable.

Residue Domination in Bounded-Treewidth Graphs

TL;DR

This work analyzes the (σ,ρ)-Dom-Set problem when σ and ρ are residue classes modulo m≥2, focusing on a treewidth parameterization. It develops a fixed-parameter algorithm running in time that solves the decision, minimization, maximization, and exact variants, by a novel DP on tree decompositions that leverages sparse languages and weight-vector compression. The authors establish matching SETH-based lower bounds for pathwidth (and thus treewidth), showing that, for the difficult residue-class pairs, no -time algorithm exists. They also address special parity cases (Lights Out variants) and prove lower bounds for Reflexive-AllOff and AllOff, complementing the upper bounds. Overall, the paper provides a near-complete picture of the fixed-parameter tractability frontier for this broad class of generalized domination problems on bounded-treewidth graphs, and offers techniques (sparse languages, defining sets, and fast joins) that may apply to other non-finite set families.

Abstract

For the vertex selection problem -DomSet one is given two fixed sets and of integers and the task is to decide whether we can select vertices of the input graph such that, for every selected vertex, the number of selected neighbors is in and, for every unselected vertex, the number of selected neighbors is in [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when and are two (potentially different) residue classes modulo . We study the problem parameterized by treewidth and present an algorithm that solves in time the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no -time algorithm parameterized by pathwidth , unless SETH fails. For , we extend these bounds to the minimization version as the decision version is efficiently solvable.
Paper Structure (23 sections, 35 theorems, 64 equations, 5 figures)

This paper contains 23 sections, 35 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Upper Bounds
  4. Lower Bounds
  5. Preliminaries
  6. Basic Notation
  7. Graphs
  8. Treewidth
  9. Partial Solutions and States
  10. Upper Bound
  11. Bounding the Size of A Single Language
  12. Compressing Weight-Vectors
  13. Faster Join Operations
  14. Dynamic Programming Algorithm
  15. Lower Bound for the Problem with Relations
  16. ...and 8 more sections

Key Result

lemma 1

Let $\sigma$ and $\rho$ denote two residue classes modulo ${\mathrm{m}} \ge 2$. Let $(G,U)$ be a graph with portals and let $L \coloneqq L(G,U) \subseteq {\mathbb A}^U$ denote its realized language. Consider two strings $x,y \in L$ with witnesses $S_x, S_y \subseteq V(G)$ such that $|S_{x} \setminus

Figures (5)

  • Figure 1: A depiction of the construction from the lower bound where ${\mathrm{m}} = 5$, $n = 4$, and $\ell = 3$.
  • Figure 2: The gadget constructions from \ref{['lem:relations:hwEqOneGeneral']}.
  • Figure 3: The gadget constructions from \ref{['lem:relations:hwOne']}. In each sketched construction we mark vertices corresponding to a feasible solution within the gadget. From the relation scope $U$, we select an arbitrary vertex.
  • Figure 4: A depiction of the clause gadget for the clause $x_1 \lor x_2 \lor \neg x_3$ as well as the negation gadget. Some indices are omitted for simplicity.
  • Figure 5: A depiction of a literal gadget and a clause gadget from the proof of the lower bound for AllOff. Some indices are omitted for simplicity.

Theorems & Definitions (83)

  • definition 1: $(\sigma,\rho)$-sets, $(\sigma,\rho)-$Dom-Set$^{}$
  • definition 2: ${\mathrm{m}}$-structured sets focke_tight_2023_lb
  • lemma 1: Compare focke_tight_2023_ub
  • lemma 1: Compare focke_tight_2023_ub
  • definition 3: Easy and Difficult Cases
  • definition 4: Lampis20
  • theorem 2: Lampis20
  • definition 5: Graph with Relations focke_tight_2023_lb
  • definition 6: $(\sigma,\rho)$-Sets of a Graph with Relations, $(\sigma,\rho)-$Dom-Set$^{\text{Rel}}$
  • lemma 1
  • ...and 73 more