Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourier meets M\"{o}bius: fast subset convolution   We present a fast algorithm for the subset convolution problem: given
functions f and g defined on the lattice of subsets of an n-element set N,
compute their subset convolution f*g, defined for all S\subseteq N by (f *
g)(S) = \sum_{T \subseteq S}f(T) g(S\setminus T), where addition and
multiplication is carried out in an arbitrary ring. Via M\"{o}bius transform
and inversion, our algorithm evaluates the subset convolution in O(n^2 2^n)
additions and multiplications, substantially improving upon the straightforward
O(3^n) algorithm. Specifically, if the input functions have an integer range
{-M,-M+1,...,M}, their subset convolution over the ordinary sum-product ring
can be computed in O^*(2^n log M) time; the notation O^* suppresses
polylogarithmic factors. Furthermore, using a standard embedding technique we
can compute the subset convolution over the max-sum or min-sum semiring in
O^*(2^n M) time. To demonstrate the applicability of fast subset convolution,
we present the first O^*(2^k n^2 + n m) algorithm for the minimum Steiner tree
problem in graphs with n vertices, k terminals, and m edges with bounded
integer weights, improving upon the O^*(3^k n + 2^k n^2 + n m) time bound of
the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent
O^*(2^n)-time algorithms for covering and partitioning problems (Bj\"{o}rklund
and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

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The categories of this paper are: eess
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph