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

2025-10-20

AI Summary

This study investigates the feasibility of transforming an infeasible initial configuration into a feasible solution for graph problems via a finite sequence of token slides to adjacent vertices—without imposing a budget constraint $b$ on the number of moves. We formalize the problem using monadic second-order logics (MSO₁, MSO₂) and first-order logic (FO), and systematically analyze its parameterized complexity with respect to structural graph parameters—including treewidth, neighborhood diversity, bandwidth, and star modularity. Our main contributions are: (i) the first meta-theorem framework for budget-free logic-defined graph transformation problems; (ii) an XP algorithm for MSO₂-Reachability parameterized by treewidth; (iii) an FPT algorithm for MSO₁-Reachability parameterized by neighborhood diversity; and (iv) W[1]-hardness results for both FO-Reachability and MSO₁-Reachability under multiple structural parameters, thereby precisely delineating the tractability frontier for budget-free reachability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Investigating meta-theorems for solution discovery in graph problems
• Analyzing tractability of MSO and FO logic problems via structural parameters
• Establishing parameterized complexity results for token sliding transformations

Innovation

• MSO2-Discovery in XP with treewidth parameterization
• MSO1-Discovery FPT with neighborhood diversity parameter
• FO-Discovery W[1]-hard with modulator parameters