NDLI: Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

Content Provider	IEEE Xplore Digital Library
Author	Curticapean, R. Mingji Xia
Copyright Year	2015
Description	Author affiliation: State Key Lab. of Comput. Sci., Inst. of Software, Beijing, China (Mingji Xia) \|\| Inst. for Comput. Sci. & Control, Budapest, Hungary (Curticapean, R.)
Abstract	We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph C. These generalize the well-known tractable planar case, and they include the genus of C, its apex number (the minimum number of vertices whose removal renders C planar), and its Hadwiger number (the size of a largest clique minor). To study these parameters, we first introduce the notion of combined matchgates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and matchgates: Using combined matchgates, we can simulate certain nonexisting gadgets F as linear combinations of L = O(1) existing gadgets. If a graph C features k occurrences of F, we can then reduce C to $t^{k}$ graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known $4^{g}n^{O(1)}$ time algorithms for PerfMatch on graphs of genus g. Orthogonally to this, we show #W[1]-hardness of the permanent on k-apex graphs, implying its ⊕W[1]-hardness under the Hadwiger number. Additionally, we rule out no(k/ log k) time algorithms under the counting exponential-time hypothesis #ETH. Finally, we use combined matchgates to prove $W[1]-hardness of evaluating the permanent modulo 2k, complementing an $O(n^{4k-3})$ time algorithm by Valiant and answering an open question of Bjϋrklund. We also obtain a lower bound of nΩ(k/ log k) under the parity version $ETH of the exponential-time hypothesis.
Starting Page	994
Ending Page	1009
File Size	379210
Page Count	16
File Format	PDF
ISBN	9781467381918
ISSN	02725428
DOI	10.1109/FOCS.2015.65
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-10-17
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Complexity theory Transmission line matrix methods Partitioning algorithms Computer science Polynomials Structural engineering Bipartite graph matchgates permanent perfect matchings parameterized counting complexity genus apex number graph minors Hadwiger number modular counting complexity Holant problem
Content Type	Text
Resource Type	Article

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Parameterizing the Permanent: Genus, Apices, Minors, Evaluation mod 2^k

Counting Problems in Parameterized Complexity

A New Direction for Counting Perfect Matchings

COUNTING PERFECT MATCHINGS

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Space complexity of perfect matching in bounded genus bipartite graphs (2010)

The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant Problems

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Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k

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Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k