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Parameterizing the Permanent: genus, apices, minors, evaluation mod 2^k 56

Parameterizing the Permanent: genus, apices, minors, evaluat...

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56th IEEE Annual Symposium on Foundations of Computer Science (FOCS)

作者： Curticapean, Radu Xia, Mingji Hungarian Acad Sci MTA SZTAKI Inst Comp Sci & Control Budapest Hungary Chinese Acad Sci Inst Software State Key Lab Comp Sci Beijing Peoples R China

ISBN: (纸本)9781467381918

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 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 n(o(k/log k)) time algorithms under the counting exponential-time hypothesis #ETH. Finally, we use combined matchgates to prove circle plus W[1]-hardness of evaluating the permanent modulo 2(k), complementing an O(n(4k-3)) time algorithm by Valiant and answering an open question of Bjorklund. We also obtain a lower bound of n(Omega(k/log k)) under the parity version circle plus ETH of the exponential-time hypothesis.

关键词： permanent perfect matchings parameterized counting complexity genus apex number graph minors Hadwiger number modular counting complexity matchgates

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Parameterizing the Permanent: genus, apices, minors, evaluation mod 2~k

Parameterizing the Permanent: genus, apices, minors, evaluat...

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IEEE Annual Symposium on Foundations of Computer Science

作者： Radu Curticapean Mingji Xia Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI) State Key Laboratory of Computer Science Institute of Software Chinese Academy of Sciences

ISBN: (纸本)9781467381925

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 t~k graphs that feature only existing gadgets, thus enabling parameterized reductions. As applications of this technique, we simplify known 4~gn~(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 n~(o(k/log k)) time algorithms under the counting exponential-time hypothesis #ETH. Finally, we use combined matchgates to prove {direct+}W[1]-hardness of evaluating the permanent modulo 2~k, complementing an O(n~(4k-3)) time algorithm by Valiant and answering an open question of Bjorklund. We also obtain a lower bound of n~(Ω(k/log k)) under the parity version {direct+}ETH of the exponential-time hypothesis.

关键词： Permanent Perfect matchings Parameterized counting complexity Genus Apex number Graph minors Hadwiger number modular counting complexity Matchgates

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