NDLI: Counting List Matrix Partitions of Graphs

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Counting List Matrix Partitions of Graphs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Goldberg, Leslie Ann McQuillan, Colin Yamakami, Tomoyuki Göbel, Andreas Richerby, David
Copyright Year	2015
Abstract	Given a symmetric $D\times D$ matrix $M$ over \0,1,*\, a list $M$-partition of a graph $G$ is a partition of the vertices of $G$ into $D$ parts which are associated with the rows of $M$. The part of each vertex is chosen from a given list in such a way that no edge of $G$ is mapped to a 0 in $M$ and no nonedge of $G$ is mapped to a 1 in $M$. Many important graph-theoretic structures can be represented as list $M$-partitions including graph colorings, split graphs, and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices $M$ computations involving list $M$-partitions are tractable. This paper focuses on the problem of counting list $M$-partitions, given a graph $G$ and given a list for each vertex of $G$. We identify a certain set of tractable matrices $M$. We give an algorithm that counts list $M$-partitions in polynomial time for every (fixed) matrix $M$ in this set. The algorithm relies on data structures such as sparse-dense partitions and subcube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of $M$ in which the interactions of 0s and 1s are controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (${\#CSP}$s), which we show how to solve using a constraint satisfaction technique known as arc-consistency. For every matrix $M$ for which our algorithm fails, we show that the problem of counting list $M$-partitions is ${\#P}$-complete. Furthermore, we give an explicit characterization of the dichotomy theorem: counting list $M$-partitions is tractable (in ${FP}$) if the matrix $M$ has a structure called a derectangularizing sequence. If $M$ has no derectangularizing sequence, we show that counting list $M$-partitions is ${\#P}$-hard. We show that the metaproblem of determining whether a given matrix has a derectangularizing sequence is ${NP}$-complete. Finally, we show that list $M$-partitions can be used to encode cardinality restrictions in $M$-partitions problems, and we use this to give a polynomial-time algorithm for counting homogeneous pairs in graphs.
Starting Page	1089
Ending Page	1118
Page Count	30
File Format	PDF
ISSN	00975397
DOI	10.1137/140963029
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4
Volume Number	44
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-08-11
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Analysis of algorithms and problem complexity ${\#P}$-completeness complexity dichotomy counting problems matrix partitions of graphs Coloring of graphs and hypergraphs graph algorithms
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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