图的列表矩阵划分的计数
Counting list matrix partitions of graphs
摘要：
给定{0，1，*}上的对称D*D矩阵M，图G的列表M-划分是将图G的顶点划分为与M的行相关联的D部分。每个顶点的部分是从给定的列表中选择的，使得G没有边映射到M中的0，G没有非边映射到M中的1。在弱完美图和强完美图的证明中，出现了许多重要的图论结构，如图的染色、分裂图、齐次集和齐次偶等，都可以表示为列表M-划分。已经有相当多的工作，以确定哪些矩阵M计算涉及列表M-分区是易处理的。我们重点研究了列表M-划分的计数问题，给定一个图G和G的每个顶点的列表。我们确定了一组“易处理”的矩阵，并给出了一个算法，计数列表M-分区在多项式时间为每个（固定）矩阵M在这个集合。该算法使用数据结构，如稀疏-密集分区和子立方体分解，以减少每个实例到一个序列的实例，其中的列表限制访问的部分M，其中0和1的相互作用受到控制。我们解决由此产生的限制性实例转换成计数约束满足问题（#CSP），我们解决使用弧一致性。对于我们的算法失败的每个矩阵M，我们证明了计数列表M-分区是#P-完全的。此外，我们给出了二分法定理的一个明确的特征：计数列表M-分区是在FP中，如果M有一个称为去矩形化序列的结构;否则，计数列表M-分区是#P-hard。我们证明了确定一个给定矩阵是否有一个去矩形化序列的元问题是NP-完全的。最后，我们表明，列表可以用来编码基数限制在M-分区问题，并使用此给一个多项式时间算法计算图中的同质对。
英文摘要：
Given a symmetric D*D matrix M over {0,1,*}, a list M-partition of a graph G is a partition of G's vertices into D parts associated with the rows of M. The part of each vertex is chosen from a given list so that no edge of G maps to a 0 in M and no non-edge of G maps to a 1 in M. Many important graph-theoretic structures can be represented as list M-partitions, such as graph colourings, split graphs and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. There has been quite a bit of work on determining for which matrices M computations involving list M-partitions are tractable. We focus on counting list M-partitions, given a graph G and a list for each vertex of G. We identify a set of "tractable" matrices and give an algorithm that counts list M-partitions in polynomial time for every (fixed) matrix M in this set. The algorithm uses data structures such as sparse-dense partitions and subcube decompositions to reduce each instance to a sequence of instances in which the lists restrict access to portions of M in which the interaction of 0s and 1s is controlled. We solve the resulting restricted instances by converting them into counting constraint satisfaction problems (#CSPs) which we solve using arc-consistency. For every matrix M for which our algorithm fails, we show that counting list M-partitions is #P-complete. Further, we give an explicit characterisation of the dichotomy theorem: counting list M-partitions is in FP if M has a structure called a derectangularising sequence; otherwise, counting list M-partitions is #P-hard. We show that the meta-problem of determining whether a given matrix has a derectangularising sequence is NP-complete. Finally, we show that lists can be used to encode cardinality restrictions in M-partitions problems and use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs.
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