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2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (2015)
Berkeley, CA, USA
Oct. 17, 2015 to Oct. 20, 2015
ISSN: 0272-5428
ISBN: 978-1-4673-8191-8
pp: 994-1009
ABSTRACT
We identify and study relevant structural parameters for the problem of counting perfect matchings in a given input graph G. These generalize the well-known tractable planar case, and they include the genus of G, its apex number (the minimum number of vertices whose removal renders G planar), and its Hadwiger number (the size of a largest clique minor). To study these parameters, we first introduce the notion of combined match gates, a general technique that bridges parameterized counting problems and the theory of so-called Holants and match gates: Using combined match gates, we can simulate certain non-existing gadgets F as linear combinations of t=O(1) existing gadgets. If a graph G features k occurrences of F, we can then reduce G to tk graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known 4g*n3 time algorithms for counting perfect matchings 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 match gates to prove parity-W[1]-hardness of evaluating the permanent modulo 2k, complementing an O(n4k) time algorithm by Valiant and answering an open question of Björklund. We also obtain a lower bound of nΩ(k/log k) under the parity version of the exponential-time hypothesis.
INDEX TERMS
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
CITATION
Radu Curticapean, Mingji Xia, "Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k", 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), vol. 00, no. , pp. 994-1009, 2015, doi:10.1109/FOCS.2015.65
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