矩阵划分的推广：复杂性与障碍
Generalisations of Matrix Partitions : Complexity and Obstructions
AI总结：
本文将矩阵划分问题推广到关系结构，证明其与三图同态P时间等价，并研究了二分性、有限障碍与有限对偶性的关系，同时证明树上的矩阵划分同态判定是NP完全的，揭示了其与CSP在可处理性上的显著差异。
摘要：
三图（trigraph）是一种图，其中每对顶点被标记为0（非弧）、1（弧）或$\star$（既是弧又是非弧）。Hell及其合作者在一系列论文中提出研究从图到三图的同态的复杂性，这类问题称为矩阵划分问题（Matrix Partition Problems），其中弧和非弧都可以映射到$\star$-弧，而非弧不能映射到弧，反之亦然。尽管矩阵划分问题是CSP的推广，但它们与CSP一样具有“本质上”组合的性质。因此，关于可能的P时间与NP完全二分性的问题是一个非常自然的问题，并在Hell等人的论文中被提出。我们提出了矩阵划分到关系结构的推广，并针对二分性问题对它们进行研究。我们首先证明三图同态和矩阵划分是P时间等价的，然后证明（就具有二分性而言）也可以限制到只有一个关系的关系结构。由于未能证明有向图上的矩阵划分与关系结构上的矩阵划分不是P时间等价的，我们通过形式化CSP情形中使用的归约，并证明这种归约不能用于矩阵划分的情形，给出了一些证据表明这不太可能。然后我们将注意力转向可以由有限组（诱导子图）障碍描述的矩阵划分。我们特别证明，任何这样的问题具有有限多个极小障碍当且仅当它具有有限对偶性。最后我们证明，在树（视为三图）上，判定给定树是否同态于另一个输入三图是NP完全的。后者显示了CSP与矩阵划分在可处理性上的显著差异，因为众所周知CSP在树类上是可处理的。
英文摘要：
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.
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