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

TL;DR

The paper develops a structural, CSP-based framework to understand the complexity of sandwich problems for matrix partitions by translating M-partitions into CSPs over reflexive complete 2-edge-coloured graphs. It provides a dichotomy theorem that either yields a tractable decomposition via homogeneous concatenations (and Datalog/bounded width) or proves NP-hardness through hereditary pp-constructs of $\mathbb{K}_3$, thereby extending the Hell–Nesetril classification to this graph category. The authors also connect these structural insights to graph sandwich problems, obtaining a P vs NP-complete dichotomy and a universal algorithm for broad matrix-partition families. The results unify CSP theory, polymorphisms, and sandwich problems, offering both a conceptual and algorithmic path toward solving general matrix-partition instances. The work also outlines open problems, including extensions to non-reflexive/complete cases and the list-version of SP for M-partitions.

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--Nešetřil 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.

Figures (8)

• Figure 1: An illustration of the operations $(\cdot)^\ast$, and $\overline{(\cdot)}$ with the structures $\mathbb K_2$, $\mathbb K_2^\ast$, and $\overline{\mathbb K_2^\ast}$.
• Figure 2: To the left, the (uncoloured) graph $\mathbb K_1 + \mathbb K_2$. In the middle and right, two $2$-edge-coloured graphs obtained from $\mathbb K_1 + \mathbb K_2$: the former by colouring edges with blue and non-edges with red, and the latter by colouring edges with red and non-edges with blue.
• Figure 3: Some minimal hereditarily-hard reflexive complete $2$-edge-coloured graphs.
• Figure 4: Two paths on three vertices such that if a reflexive complete $2$-edge-coloured graph $\mathbb H$ contains either of them as subgraphs, then $\mathbb H$ pp-constructs $\mathbb K_3$.
• Figure 5: A $2$-edge-coloured graph $\mathbb H_3$ on three vertices that is hereditarily hard for the class of reflexive $2$-edge-coloured graphs.

Theorems & Definitions (76)

• Theorem 3
• Theorem 4: wonderland
• Example 1
• Theorem 5: wonderland
• Theorem 6
• Corollary 7
• proof
• Lemma 11
• proof
• Lemma 12