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arXiv · 2510.17344

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

Nicolas Bousquet·Amer E. Mouawad·Stephanie Maaz·Naomi Nishimura·Sebastian Siebertz
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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.

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BibTeXRIS

Nicolas Bousquet, Amer E. Mouawad, Stephanie Maaz, Naomi Nishimura, Sebastian Siebertz. 2025-10-20. On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers. https://arxiv.org/abs/2510.17344

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