Generalisations of Matrix Partitions : Complexity and Obstructions

TL;DR

The article generalizes Matrix Partition Problems (MP) from graphs to trigraphs and relational signatures, and studies when a dichotomy between $P$ and $NP$-complete problems holds. It establishes a deterministic $P$-time equivalence between MP and its star variant $MP_\star$, and shows that MP_∅ is $P$-time reducible to a CSP instance, enabling a CSP-style dichotomy for that variant; it also proves arity-reduction results that connect MP across signatures of different shapes. The work further develops obstruction theory, proving that having finitely many obstructions is equivalent to finite duality for MP and MP_\star, while providing examples and counterexamples that illuminate the limits of these correspondences. It demonstrates NP-hardness for MP on trees via a 3-SAT reduction, highlighting a notable tractability gap with classical CSP results on trees. The paper thereby lays a framework linking MP, CSP, obstructions, and tractability, and raises open questions about the full equivalence of MP on directed graphs with MP on general finite signatures and the complete dichotomy landscape for MP.

Abstract

A trigraph is a graph where each pair of vertices is labelled either 0 (a non-arc), 1 (an arc) or $\star$ (both an arc and a non-arc). In a series of papers, Hell and co-authors proposed to study the complexity of homomorphisms from graphs to trigraphs, called Matrix Partition Problems, where arcs and non-arcs can be both mapped to $\star$-arcs, while a non-arc cannot be mapped to an arc, and vice-versa. Even though Matrix Partition Problems are generalisations of CSPs, they share with them the property of being ``intrinsically'' combinatorial. So, the question of a possible P-time vs NP-complete dichotomy is a very natural one and was raised in Hell et al.'s papers. We propose a generalisation of Matrix Partitions to relational structures and study them with respect to the question of a dichotomy. We first show that trigraph homomorphisms and Matrix Partitions are P-time equivalent, and then prove that one can also restrict (with respect to having a dichotomy) to relational structures with a single relation. Failing in proving that Matrix Partitions on directed graphs are not P-time equivalent to Matrix Partitions on relational structures, we give some evidence that it might be unlikely by formalising the reductions used in the case of CSPs and by showing that such reductions cannot work for the case of Matrix Partitions. We then turn our attention to Matrix Partitions that can be described by finite sets of (induced-subgraph) obstructions. We show, in particular, that any such problem has finitely many minimal obstructions if and only if it has finite duality. We conclude by showing that on trees (seen as trigraphs) it is NP-complete to decide whether a given tree has a homomorphism to another input trigraph. The latter shows a notable difference on tractability between CSP and Matrix Partitions as it is well-known that CSP is tractable on the class of trees.

Figures (7)

• Figure 1: Hasse diagrams of the four posets.
• Figure 2: Symmetric $\star$-graphs from Examples \ref{['ex:one']} and \ref{['ex:two']}. 1-edges are thick, $\star$-edges are thin.
• Figure 3: Dichotomy implications. Each arrow shows an implication of the existence of a dichotomy, i.e., if the class at the tail has a dichotomy, then the class at the head has it. The vertical ones are shown in \ref{['sec:mpstar=mp']}, and the horizontal ones are shown in \ref{['sec:arity']}.
• Figure 4: 01-trees that represent the negation types.
• Figure 5: Correspondence between paths and variables.

Theorems & Definitions (66)

• Definition 2.1: $({*},\sigma)$-structures
• Definition 2.2: homomorphism for $({*},\sigma)$-structures
• Remark
• Proposition 2.3
• Definition 2.4: Generalised Matrix Partition
• Remark
• Example 1
• Example 2
• Remark
• Theorem 3.2