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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
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.
Complexity theory, Transmission line matrix methods, Partitioning algorithms, Computer science, Polynomials, Structural engineering, Bipartite graph

R. Curticapean and M. Xia, "Parameterizing the Permanent: Genus, Apices, Minors, Evaluation Mod 2k," 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, CA, USA, 2015, pp. 994-1009.
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