On the complexity of Sandwich Problems for $M$-partitions

2026-02-10

AI Summary

This work investigates the computational complexity classification of constraint satisfaction problems (CSPs) defined by reflexive complete 2-edge-colored graphs, with a focus on matrix partitioning problems and their sandwich variants. By extending the Hell–Nešetřil homomorphism dichotomy theorem to this generalized framework and integrating tools from graph theory, algebraic structures, and complexity analysis, the paper presents the first efficient algorithm applicable to general matrix partitioning problems. It establishes a complete and polynomial-time verifiable dichotomy for the sandwich problem, precisely delineating instances that are solvable in polynomial time from those that are NP-complete, thereby providing a clear computational boundary for this class of problems.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSecurity and Privacy: Applications of cryptography

Abstract

We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell--Ne\v{s}et\v{r}il theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

Problem

• Sandwich Problems
• M-partitions
• Constraint Satisfaction Problems
• Computational Complexity
• Matrix Partition

Innovation

• constraint satisfaction problem
• matrix partition
• sandwich problem
• dichotomy
• 2-edge-coloured graph