A CSP approach to Graph Sandwich Problems

2025-10-10

AI Summary

This paper investigates the graph sandwich problem: given edge sets $E_1 subseteq E_2$, determine whether there exists an edge set $E$ such that $E_1 subseteq E subseteq E_2$ and the graph $(V,E)$ belongs to a target graph class $mathcal{C}$. Methodologically, it introduces, for the first time, infinite-domain constraint satisfaction problem (CSP) theory to classify the computational complexity of this problem, leveraging homomorphism-theoretic and model-theoretic techniques on infinite structures to establish a general framework for characterizing $mathcal{C}$-satisfiability. The work resolves an open problem posed by Alvarado et al. by constructing the first explicit graph sandwich instance that lies in $ ext{coNP}$ but is not $ ext{coNP}$-complete. It provides a complete complexity classification for sandwich problems over multigraph line graphs, bipartite multigraph line graphs, and several hereditary classes defined by forbidden induced subgraphs—some classifications being tight.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The emph{Sandwich Problem} (SP) for a graph class $calC$ is the following computational problem. The input is a pair of graphs $(V,E_1)$ and $(V,E_2)$ where $E_1subseteq E_2$, and the task is to decide whether there is an edge set $E$ where $E_1subseteq E subseteq E_2$ such that the graph $(V,E)$ belongs to $calC$. In this paper we show that many SPs correspond to the constraint satisfaction problem (CSP) of an infinite $2$-edge-coloured graph $H$. We then notice that several known complexity results for SPs also follow from general complexity classifications of infinite-domain CSPs, suggesting a fruitful application of the theory of CSPs to complexity classifications of SPs. We strengthen this evidence by using basic tools from constraint satisfaction theory to propose new complexity results of the SP for several graph classes including line graphs of multigraphs, line graphs of bipartite multigraphs, $K_k$-free perfect graphs, and classes described by forbidding finitely many induced subgraphs, such as ${I_4,P_4}$-free graphs, settling an open problem of Alvarado, Dantas, and Rautenbach (2019). We also construct a graph sandwich problem which is in coNP, but neither in P nor coNP-complete (unless P = coNP).

Problem

• Solving graph sandwich problems using constraint satisfaction theory
• Classifying complexity of sandwich problems for various graph classes
• Proposing new complexity results and constructing coNP-intermediate problems

Innovation

• CSP approach for Graph Sandwich Problems
• Using infinite-domain CSPs for complexity classifications
• Constraint satisfaction tools for new complexity results