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

TL;DR

This paper classifies the complexity of certain constraint satisfaction problems related to reflexive complete 2-edge-coloured graphs, extending known graph homomorphism results and providing efficient algorithms for specific classes.

Contribution

It offers a polynomial-time classification of CSPs for reflexive complete 2-edge-coloured graphs and applies this to matrix partition and sandwich problems, including a P vs. NP-complete dichotomy.

Findings

Polynomial-time checkability of CSP complexity for reflexive complete 2-edge-coloured graphs

Extension of Hell–Nešetřil theorem to this class of graphs

Classification of sandwich problems for matrix partitions as P or NP-complete

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.