On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers

TL;DR

This paper explores the complexity of solution discovery in graph problems defined by logical formulas, establishing both tractability results and hardness barriers based on structural graph parameters.

Contribution

It provides new fixed-parameter tractability and XP results for MSO and FO solution discovery problems, along with hardness results showing limits of algorithmic approaches.

Findings

MSO$_2$-Discovery is in XP when parameterized by treewidth.

MSO$_1$-Discovery is fixed-parameter tractable with neighborhood diversity.

FO-Discovery is W[1]-hard with certain graph parameters.

Abstract

Solution discovery asks whether a given (infeasible) starting configuration to a problem can be transformed into a feasible solution using a limited number of transformation steps. This paper investigates meta-theorems for solution discovery for graph problems definable in monadic second-order logic (MSO$_1$ and MSO$_2$) and first-order logic (FO) where the transformation step is to slide a token to an adjacent vertex, focusing on parameterized complexity and structural graph parameters that do not involve the transformation budget $b$. We present both positive and negative results. On the algorithmic side, we prove that MSO$_2$-Discovery is in XP when parameterized by treewidth and that MSO$_1$-Discovery is fixed-parameter tractable when parameterized by neighborhood diversity. On the hardness side, we establish that FO-Discovery is W[1]-hard when parameterized by modulator to stars, modulator to paths, as well as twin cover, numbers. Additionally, we prove that MSO$_1$-Discovery is W[1]-hard when parameterized by bandwidth. These results complement the straightforward observation that solution discovery for the studied problems is fixed-parameter tractable when the budget $b$ is included in the parameter (in particular, parameterized by cliquewidth$+b$, where the cliquewidth of a graph is at most any of the studied parameters), and provide a near-complete (fixed-parameter tractability) meta-theorems investigation for solution discovery problems for MSO- and FO-definable graph problems and structural parameters larger than cliquewidth.