We consider the weighted satisfiability problem for Boolean circuits and propositional formule, where the weight of an assignment is the number of variables set to true. We study the parameterizedcomplexity of these ...
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We consider the weighted satisfiability problem for Boolean circuits and propositional formule, where the weight of an assignment is the number of variables set to true. We study the parameterizedcomplexity of these problems and initiate a systematic study of the complexity of its fragments. Only the monotone fragment has been considered so far and proven to be of same complexity as the unrestricted problems. Here, we consider all fragments obtained by semantically restricting circuits or formule to contain only gates (connectives) from a fixed set B of Boolean functions. We obtain a dichotomy result by showing that for each such B, the weighted satisfiability problems are either W [P] -complete (for circuits) or W [SAT] -complete (for formule) or efficiently solvable. We also consider the related enumeration and counting problems.
The classes W[P] and W[1] are parameterized analogues of NP in that they can be characterized by machines with restricted existential nondeterminism. These machine characterizations give rise to two natural notions of...
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The classes W[P] and W[1] are parameterized analogues of NP in that they can be characterized by machines with restricted existential nondeterminism. These machine characterizations give rise to two natural notions of parameterized randomized algorithms that we call W[P]-randomization and W[1]-randomization. This paper develops the corresponding theory.
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 ...
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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 parameterizedcounting 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.
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 ...
详细信息
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 parameterizedcounting 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.
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