{"ID":2852667,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.17344","arxiv_id":"2510.17344","title":"On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers","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.","short_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...","url_abs":"https://arxiv.org/abs/2510.17344","url_pdf":"https://arxiv.org/pdf/2510.17344v1","authors":"[\"Nicolas Bousquet\",\"Amer E. Mouawad\",\"Stephanie Maaz\",\"Naomi Nishimura\",\"Sebastian Siebertz\"]","published":"2025-10-20T09:42:47Z","proceeding":"cs.DS","tasks":"[\"cs.DS\"]","methods":"[]","has_code":false}