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Fourier meets Möbius: fast subset convolution
 Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an nelement set N, compute their subset convolution f ∗g, defined for all S ⊆ N by (f ∗ g)(S) = X f(T)g(S \ T), T ⊆S where addition and multiplication is carried out in an a ..."
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Cited by 76 (10 self)
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We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an nelement set N, compute their subset convolution f ∗g, defined for all S ⊆ N by (f ∗ g)(S) = X f(T)g(S \ T), T ⊆S where addition and multiplication is carried out in an arbitrary ring. Via Möbius 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 Õ(2 n log M) time; the notation Õ suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max–sum or min–sum semiring in Õ(2n M) time. To demonstrate the applicability of fast subset convolution, we present the first Õ(2k n 2 + nm) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the Õ(3k n+2 k n 2 +nm) time bound of the classical Dreyfus– Wagner algorithm. We also discuss extensions to recent Õ(2 n)time algorithms for covering and partitioning problems
Set partitioning via inclusionexclusion
 SIAM J. Comput
"... Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of t ..."
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Cited by 59 (7 self)
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Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion–exclusion and the zeta transform. In effect we get exact algorithms in 2nnO(1) time for several wellstudied partition problems including Domatic Number, Chromatic Number, Maximum kCut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to modelbased data clustering. If only polynomial space is available, our algorithms run in time 3nnO(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461n) time and Domatic Number in O(2.8718n) time. Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and d(1 + )χ(G)e in time O(1.2209n + 2.2461e−n). 1. Introduction. Graph colouring, domatic partitioning, weighted kcut, and a
The complexity of class polynomial computation via floating point approximations. ArXiv preprint
, 601
"... Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest ..."
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Cited by 44 (7 self)
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Abstract. We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmeticgeometric mean. Under the heuristic assumption, justified by experiments, that the correctness of the result is not perturbed by rounding errors, the algorithm runs in time “p “p ”” 3 2 O Dlog D  M Dlog D  ⊆ O ` Dlog 6+ε D  ´ ⊆ O ` h 2+ε´ for any ε> 0, where D is the CM discriminant, h is the degree of the class polynomial and M(n) is the time needed to multiply two nbit numbers. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials. 1. Motivation and
Algorithms for propositional model counting.
 In Proc. of the 14th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’07),
, 2007
"... Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comp ..."
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Cited by 29 (10 self)
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Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worstcase time and space requirements.
Computing modular polynomials in quasilinear time
 Mathematics of Computation
"... Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation ..."
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Cited by 27 (5 self)
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Abstract. We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation of modular functions and on interpolation has a complexity that is up to logarithmic factors linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomial Φℓ of prime level ℓ in time O ( ℓ 2 log 3 ℓM(ℓ) ) ⊆ O ( ℓ 3 log 4+ε ℓ), where M(ℓ) is the time needed to multiply two ℓbit numbers. Besides treating modular polynomials for Γ0 (ℓ), which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schläfli polynomials and their generalisations.
Efficient authentication from hard learning problems
 EUROCRYPT
"... Abstract. We construct efficient authentication protocols and messageauthentication codes (MACs) whose security can be reduced to the learning parity with noise (LPN) problem. Despite a large body of work – starting with the HB protocol of Hopper and Blum in 2001 – until now it was not even known ho ..."
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Cited by 21 (6 self)
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Abstract. We construct efficient authentication protocols and messageauthentication codes (MACs) whose security can be reduced to the learning parity with noise (LPN) problem. Despite a large body of work – starting with the HB protocol of Hopper and Blum in 2001 – until now it was not even known how to construct an efficient authentication protocol from LPN which is secure against maninthemiddle (MIM) attacks. A MAC implies such a (tworound) protocol. 1
Dynamic programming on tree decompositions using generalised fast subset convolution
 Proceedings of the 17th Annual European Symposium on Algorithms, ESA 2009
"... Abstract. In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk23k) tim ..."
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Cited by 20 (1 self)
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Abstract. In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk23k) time and the problem to count the number of perfect matchings in O∗(2k) time. Using a generalisation of fast subset convolution, we obtain faster algorithms for all [ρ, σ]domination problems with finite or cofinite ρ and σ on tree decompositions. These include many well known graph problems. We give additional results on many more graph covering and partitioning problems. 1
Saving Space by Algebraization
, 2010
"... The Subset Sum and Knapsack problems are fundamental N Pcomplete problems and the pseudopolynomial time dynamic programming algorithms for them appear in every algorithms textbook. The algorithms require pseudopolynomial time and space. Since we do not expect polynomial time algorithms for Subset ..."
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Cited by 13 (2 self)
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The Subset Sum and Knapsack problems are fundamental N Pcomplete problems and the pseudopolynomial time dynamic programming algorithms for them appear in every algorithms textbook. The algorithms require pseudopolynomial time and space. Since we do not expect polynomial time algorithms for Subset Sum and Knapsack to exist, a very natural question is whether they can be solved in pseudopolynomial time and polynomial space. In this paper we answer this question affirmatively, and give the first pseudopolynomial time, polynomial space algorithms for these problems. Our approach is based on algebraic methods and turns out to be useful for several other problems as well. Then we show how the framework yields polynomial space exact algorithms for the classical Traveling Salesman, Weighted Set Cover and Weighted Steiner Tree problems as well. Our algorithms match the time bound of the best known pseudopolynomial space algorithms for these problems.
Algorithms for monitoring realtime properties
 In Proceedings of the 2nd International Conference on Runtime Verification, volume 7186 of LNCS
, 2012
"... Abstract. We present and analyze monitoring algorithms for a safety fragment of metric temporal logics, which differ in their underlying time model. The time models considered have either dense or discrete time domains and are pointbased or intervalbased. Our analysis reveals differences and simi ..."
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Cited by 12 (4 self)
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Abstract. We present and analyze monitoring algorithms for a safety fragment of metric temporal logics, which differ in their underlying time model. The time models considered have either dense or discrete time domains and are pointbased or intervalbased. Our analysis reveals differences and similarities between the time models for monitoring and highlights key concepts underlying our and prior monitoring algorithms. 1