Computational complexity of counting problems on 3-regular planar graphs

A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial in...

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Bibliographic Details
Published in:Theoretical computer science Vol. 384; no. 1; pp. 111 - 125
Main Authors: Xia, Mingji, Zhang, Peng, Zhao, Wenbo
Format: Journal Article Conference Proceeding
Language:English
Published: Amsterdam Elsevier B.V 24.09.2007
Elsevier
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
Exact sciences and technology
Holographic reduction
Information retrieval. Graph
Matching
Mathematics
Miscellaneous
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
P-completeness
Sciences and techniques of general use
Theoretical computing
Vertex cover
Matching
Vertex cover
Holographic reduction
P-completeness
Polynomial
Recurrence
Vertex
Computer theory
Recurrence relation
Planar graph
Computational complexity
Sufficient condition
Completeness
Polynomial interpolation
Application
Variety
Counting
Regular graph
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary:A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial interpolation technique in high dimension to prove the #P-completeness of problems on graphs with special requirements. Using this method, we show that #3-Regular Bipartite Planar Vertex Covers is #P-complete. Furthermore, we use Valiant’s Holant Theorem to construct a holographic reduction from it to #2,3-Regular Bipartite Planar Matchings, establishing the #P-completeness of the latter. Finally, we completely classify the problems #Planar Read-twice 3SAT with different ternary symmetric relations according to their computational complexity, by giving several more applications of holographic reduction in proving the #P-completeness of the corresponding counting problems.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2007.05.023