Kernelization Complexity of Solution Discovery Problems

TL;DR

This work analyzes kernelization complexity for solution-discovery variants of several classic graph problems under token sliding. By introducing the MMO auxiliary problem and a suite of gadgets, it systematically derives both positive kernelization results (polynomial kernels in $k$ for IS-D, VC-D, DS-D, and Mat-D on relevant graph classes) and tight kernel lower bounds (no polynomial kernel in $b+pw$ for VC-D and IS-D; no polynomial kernel in $k$ for SP-D and VCut-D under standard assumptions). The authors rely on log-space reductions, pl-reductions, and or-cross-compositions to transfer hardness from MMO to target problems and to prove kernel-size intractability, while leveraging nowhere-dense graph properties and domination-core techniques for positive results. The work thus provides a comprehensive, structurally nuanced classification of kernelization possibilities across a broad set of solution-discovery problems in token-sliding models, with implications for both theory and practical reconfiguration tasks.

Abstract

In the solution discovery variant of a vertex (edge) subset problem $Π$ on graphs, we are given an initial configuration of tokens on the vertices (edges) of an input graph $G$ together with a budget $b$. The question is whether we can transform this configuration into a feasible solution of $Π$ on $G$ with at most $b$ modification steps. We consider the token sliding variant of the solution discovery framework, where each modification step consists of sliding a token to an adjacent vertex (edge). The framework of solution discovery was recently introduced by Fellows et al. [Fellows et al., ECAI 2023] and for many solution discovery problems the classical as well as the parameterized complexity has been established. In this work, we study the kernelization complexity of the solution discovery variants of Vertex Cover, Independent Set, Dominating Set, Shortest Path, Matching, and Vertex Cut with respect to the parameters number of tokens $k$, discovery budget $b$, as well as structural parameters such as pathwidth.

Figures (17)

• Figure 1: A classification of problems into three categories: (blue, alternatively grid) problems for which we obtain polynomial kernels, (white) those for which polynomial kernels are unlikely, and (red, alternatively lines) those for which fixed-parameter tractable algorithms are unlikely. Each entry in a category mentions a solution discovery problem, one or more parameters (in parentheses and followed by a dash), and the graph class with respect to which the problem falls into the category. A reference in the parentheses indicates that the fixed-parameter tractability of that problem was established in the cited work. $pw$ denotes the pathwidth and fvs denotes the feedback vertex set number of the input graph.
• Figure 2: Graph classes considered in this paper. Arrows indicate inclusion.
• Figure 3: An MMO-edge-$e$$G_e$ for an edge $uv = e \in E(H)$ for a graph $H$ and edge weight function $\sigma$ of an MMO instance, with $\sigma(e) = 4$.
• Figure 4: An MMO-vertex-$v$$G_v$ for a vertex $v \in V(H)$ for a graph $H$, edge weight function $\sigma$, and integer $r$ of an MMO instance. The vertex $v$ is incident to edges $e_1, e_2 \in E(H)$, $\sigma(e_1) = 3$, $\sigma(e_2) = 2$, and $r= 4$.
• Figure 5: Edges from one MMO-edge-$e$, for an edge $e = uv$ for a graph $H$, edge weight function $\sigma$, and integer $r$ of an MMO instance, to the MMO-vertex-$u$ and MMO-vertex-$v$ subgraphs in $G_H$. Red is used for edges between vertices in $B$ and $Z$ and yellow is used for edges between vertices in $\{e^u, e^v\}$ and $Y$. $\sigma(e) = 2$ and $r = 4$.

Theorems & Definitions (95)

• Definition 2.1
• Definition 2.2
• Theorem 2.1: kreutzer2018polynomialpilipczuk2018number
• Definition 2.3
• Definition 2.4
• Definition 2.5: bodlaender2014kernelization
• Theorem 2.2: bodlaender2014kernelization
• Lemma 3.1
• proof
• Claim 1