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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 tk graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known 4gnO(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 &#x2295;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(n4k-3) time algorithm by Valiant and answering an open question of Bj&#x03C5;&#x0308;rklund. We also obtain a lower bound of n&#x03A9;(k/ log k) under the parity version $ETH of the exponential-time hypothesis.

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