Results 1  10
of
22
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 ..."
Abstract

Cited by 59 (7 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
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,
Fast polynomialspace algorithms using Möbius inversion: Improving on Steiner Tree and related problems
"... 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 Pcomplet ..."
Abstract

Cited by 30 (3 self)
 Add to MetaCart
(Show Context)
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.
Exact structure discovery in Bayesian networks with less space
 In Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI
, 2009
"... The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding ..."
Abstract

Cited by 29 (7 self)
 Add to MetaCart
The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding an optimal network structure. When little space is available, we apply the Gurevich– Shelah recurrence—originally proposed for the Hamiltonian path problem—and obtain time 2 2n−s n O(1) in space 2 s n O(1) for any s = n/2,n/4,n/8,...; we assume the indegree of each node is bounded by a constant. For the more practical setting with moderate amounts of space, we present a novel scheme. It yields running time 2 n (3/2) p n O(1) in space 2 n (3/4) p n O(1) for any p = 0,1,...,n/2; these bounds hold as long as the indegrees are at most 0.238n. Furthermore, the latter scheme allows easy and efficient parallelization beyond previous algorithms. We also explore empirically the potential of the presented techniques. 1
An O∗(2 n ) algorithm for graph coloring and other partitioning problems via inclusionexclusion
 IN PROCEEDINGS OF THE 47TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2006), IEEE
, 2006
"... We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partit ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
(Show Context)
We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partitioning problems, such as graph coloring, domatic partitioning, and MAX kCUT, aswell as machine learning problems like decision graph learning and modelbased data clustering. Our algorithm runs in O ∗ (2 n) time, thus substantially improving on the usual O ∗ (3 n)time dynamic programming algorithm; the notation O ∗ suppresses factors polynomial in n. This result improves, e.g., Byskov’s recent record for graph coloring from O ∗ (2.4023 n) to O ∗ (2 n). We note that twenty five years ago, R. M. Karp used inclusion–exclusion in a similar fashion to reduce the space requirement of the usual dynamic programming algorithms from exponential to polynomial.
Combinatorial bounds via measure and conquer: Boundings minimal dominating sets and applications
 PRELIM.VERSION IN PROC. 16TH ISAAC
, 2006
"... We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) boun ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) bound. Our result makes use of the measure and conquer technique which was recently developed in the area of exact algorithms. Based on this result, we derive an O(2.8718n) algorithm for the domatic number problem.
Counting subgraphs via homomorphisms
 In Automata, Languages and Programming: ThirtySixth International Colloquium (ICALP
, 2009
"... We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well known results in algorithms and combinatorics including the recent algorithm of Björklund, Husfeldt and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn, Gottlieb, Kohn, and Karp for counting Hamiltonian cycles, Ryser’s formula for counting perfect matchings of a bipartite graph, and color coding based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions and the color coding technique, we obtain the following new results: • The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time O(2 n+o(n)); in particular in time O(2 n n 3) for trees and in time 2 n+O( √ n) for planar graphs. • Counting all maximum planar subgraphs, subgraphs of bounded genus, or more generally
Exponentialtime approximation of weighted set cover
 Inf. Process. Lett
"... The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for problems of that kind. The goal is getting the time complexity still of order O(cn), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes — tradeoffs between approximation factor and the time complexity. We use general transformations from exponentialtime exact algorithms to approximations that are faster but still exponentialtime. For example, we show that for any reduction rate r, one can transform any O∗(cn)time1 algorithm for Set Cover into a (1+ln r)approximation algorithm running in time O∗(cn/r). We believe that results of that kind extend the applicability of exact algorithms for NPhard problems.
Counting Perfect Matchings as Fast as Ryser
, 2012
"... We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5) ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
We show that there is a polynomial space algorithm that counts the number of perfect matchings in an nvertex graph in O ∗ (2 n/2) ⊂ O(1.415 n) time. (O ∗ (f(n)) suppresses functions polylogarithmic in f(n)).The previously fastest algorithms for the problem was the exponential space O ∗ (((1 + √ 5)/2) n) ⊂ O(1.619 n) time algorithm by Koivisto, and for polynomial space, the O(1.942 n) time algorithm by Nederlof. Our new algorithm’s runtime matches up to polynomial factors that of Ryser’s 1963 algorithm for bipartite graphs. We present our algorithm in the more general setting of computing the hafnian over an arbitrary ring, analogously to Ryser’s algorithm for permanent computation. We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an n element ground set cannot be solved in O ∗ (2 (1−ɛ1)n) time for any ɛ1> 0 unless there are O ∗ (2 (1−ɛ2)n) time algorithms for computing an n × n 0/1 matrix permanent, for some ɛ2> 0 depending only on ɛ1.
TRIMMED MOEBIUS INVERSION AND GRAPHS OF BOUNDED DEGREE
"... We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an nele ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
We study ways to expedite Yates’s algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an nelement universe U and a family F of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of U with k sets from F in time within a polynomial factor (in n) of the number of supersets of the members of F. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to wellstudied combinatorial optimisation problems on graphs of maximum degree ∆. In particular, we show how to compute the Domatic Number in time within a polynomial factor of (2 ∆+1 − 2) n/(∆+1) and the Chromatic Number in time within a polynomial factor of (2 ∆+1 − ∆ − 1) n/(∆+1). For any constant ∆, these bounds are O ` (2 − ɛ) n ´ for ɛ> 0 independent of the number of vertices n.