2015/11/07 by Radu Curticapean, Mingji Xia, Curticapean, Radu +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #cs.CC
paper · pdf · doi:10.48550/arxiv.1511.02321
35 pages, appears in FOCS 2015
arxiv created 2015/11/07 · openalex publication_date 2015/11/07 · arxiv updated 2015/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We identify and study relevant structural parameters for the problem PerfMatch 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 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 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 nO(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 parity-W[1]-hardness of evaluating the permanent modulo 2k, complementing an O(n4k-3) 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.