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Supervisor
"... I would like to thank Professor Rasmus Pagh for these years of research during which he has been a precious guide. If I have gained any skill as a researcher during this period in time, it is because of his lessons and his advices. A special thank goes to Professor Gerth Brodal, Professor Thore Husf ..."
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I would like to thank Professor Rasmus Pagh for these years of research during which he has been a precious guide. If I have gained any skill as a researcher during this period in time, it is because of his lessons and his advices. A special thank goes to Professor Gerth Brodal, Professor Thore
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 80 (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
InclusionExclusion Algorithms for Counting Set Partitions
, 2006
"... Given an nelement set U and a family of subsets S ⊆ 2 U we show how to count the number of kpartitions S1 ∪ · · · ∪ Sk = U into subsets Si ∈ S in time 2 n n O(1). The only assumption on S is that it can be enumerated in time 2 n n O(1). In effect we get exact algorithms in time 2 n n O(1) fo ..."
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Cited by 36 (1 self)
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Given an nelement set U and a family of subsets S ⊆ 2 U we show how to count the number of kpartitions S1 ∪ · · · ∪ Sk = U into subsets Si ∈ S in time 2 n n O(1). The only assumption on S is that it can be enumerated in time 2 n n O(1). In effect we get exact algorithms in time 2 n n O(1) for a number of wellstudied partition problems including Domatic Number, Chromatic Number,
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 60 (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
INCLUSION–EXCLUSION BASED ALGORITHMS FOR
"... Abstract. We present a deterministic algorithm producing the number of kcolourings of a graph on n vertices in time 2nnO(1). We also show that the chromatic number can be found by a polynomial space algorithm running in time O(2.2461n). Finally, we present a family of polynomial space approximatio ..."
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Abstract. We present a deterministic algorithm producing the number of kcolourings of a graph on n vertices in time 2nnO(1). We also show that the chromatic number can be found by a polynomial space algorithm running in time O(2.2461n). Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and (1 + )χ(G) in time
Exact algorithms for exact satisfiability and number of perfect matchings
 In Proc. 33rd ICALP
, 2006
"... Abstract. We present exact algorithms with exponential running times for variants of nelement set cover problems, based on divideandconquer and on inclusion–exclusion characterisations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m l O(1) and pol ..."
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Cited by 25 (8 self)
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Abstract. We present exact algorithms with exponential running times for variants of nelement set cover problems, based on divideandconquer and on inclusion–exclusion characterisations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m l O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an nvertex graph in time 2 n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 n) and exponential space. Using the same techniques we show how to compute Chromatic Number of an nvertex graph in time O(2.4423 n) and polynomial space, or time O(2.3236 n) and exponential space. 1
A Communication Complexity Proof that Symmetric Functions have Logarithmic Depth
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS FTP:http://www.brics.aau.dk/BRICS/ ..."
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Cited by 1 (0 self)
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS FTP:http://www.brics.aau.dk/BRICS/
Finding a Path of Superlogarithmic Length
, 2002
"... We consider the problem of finding a long, simple path in an undirected graph. We present a polynomialtime algorithm that finds a path of , where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O for the Longest Path problem, where V denotes the ..."
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Cited by 19 (2 self)
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We consider the problem of finding a long, simple path in an undirected graph. We present a polynomialtime algorithm that finds a path of , where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O for the Longest Path problem, where V denotes the graph's vertices.
On evaluation of permanents
, 2009
"... The permanent of an m × n matrix A = (aij), with m ≤ n, is defined as per A a1σ(1) a2σ(2) · · · amσ(m), where the summation is over all injections σ from M σ. = {1, 2,..., m} to N. = {1, 2,..., n}. While studies on permanents – since their introduction in 1812 by Binet [3] and Cauchy [5] – have ..."
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Cited by 1 (1 self)
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The permanent of an m × n matrix A = (aij), with m ≤ n, is defined as per A a1σ(1) a2σ(2) · · · amσ(m), where the summation is over all injections σ from M σ. = {1, 2,..., m} to N. = {1, 2,..., n}. While studies on permanents – since their introduction in 1812 by Binet [3] and Cauchy [5] – have focused on matrices over fields and commutative rings, we generally only assume the entries are from some semiring, that is, multiplication need not commute and additive inverses need not exist. In this note, we give simple algorithms to evaluate the permanent of a given matrix. In arbitrary semirings, we apply Bellman–Held–Karp type
Edited in cooperation with Dominik Scheder 1 Executive Summary
"... This report documents the program and the outcomes of Dagstuhl Seminar 13331 “Exponential Algorithms: Algorithms and Complexity Beyond Polynomial Time”. Problems are often solved in practice by algorithms with worstcase exponential time complexity. It is of interest to find the fastest algorithms f ..."
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This report documents the program and the outcomes of Dagstuhl Seminar 13331 “Exponential Algorithms: Algorithms and Complexity Beyond Polynomial Time”. Problems are often solved in practice by algorithms with worstcase exponential time complexity. It is of interest to find the fastest algorithms for a given problem, be it polynomial, exponential, or something in between. The focus of the seminar is on finergrained notions of complexity than NPcompleteness and on understanding the exact complexities of problems. The report provides a rationale for the workshop and chronicles the presentations at the workshop. The report notes the progress on the open problems posed at the past workshops on the same topic. It also reports a collection of results that cite the presentations at the previous seminar. The docoument presents the collection of the abstracts of the results presented at the seminar. It also presents a compendium of open problems.
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