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Fast polynomialspace algorithms using Möbius inversion: Improving on Steiner Tree and related problems
"... Abstract. Given a graph with n vertices, k terminals and bounded integer weights on the edges, we compute the minimum Steiner Tree in O ∗ (2 k) time and polynomial space, where the O ∗ notation omits poly(n, k) factors. Among our results are also polynomialspace O ∗ (2 n) algorithms for several N P ..."
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Abstract. Given a graph with n vertices, k terminals and bounded integer weights on the edges, we compute the minimum Steiner Tree in O ∗ (2 k) time and polynomial space, where the O ∗ notation omits poly(n, k) factors. Among our results are also polynomialspace O ∗ (2 n) algorithms for several N Pcomplete spanning tree and partition problems. The previous fastest known algorithms for these problems use the technique of dynamic programming among subsets, and require exponential space. We introduce the concept of branching walks and extend the InclusionExclusion algorithm of Karp for counting Hamiltonian paths. Moreover, we show that our algorithms can also be obtained by applying Möbius inversion on the recurrences used for the dynamic programming algorithms. 1
Bayesian structure discovery in Bayesian networks with less space
"... Current exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space within a polynomial factor of 2 n. For practical use, the space requirement is the bottleneck, which motivates trading space against time. Here, previous results on finding an optimal n ..."
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Current exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space within a polynomial factor of 2 n. For practical use, the space requirement is the bottleneck, which motivates trading space against time. Here, previous results on finding an optimal network structure in less space are extended in two directions. First, we consider the problem of computing the posterior probability of a given arc set. Second, we operate with the general partial order framework and its specialization to bucket orders, introduced recently for related permutation problems. The main technical contribution is the development of a fast algorithm for a novel zeta transform variant, which may be of independent interest. 1