An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion

Koivisto, Mikko

doi:10.1109/focs.2006.11

Cited by 62 publications

(53 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, the present technique can be seen as a generalization of the authors' previous work [3,16] based on inclusion-exclusion and applies to a wider family of partitioning problems. For example, we can now solve extended partitioning problems, such as finding all k-colorable induced subgraphs of a given n-vertex graph, inÕ(2 n ) total time.…”

Section: Application To Specific Computational Problemsmentioning

confidence: 92%

Fourier meets möbius: fast subset convolution

Björklund

Husfeldt

Kaski

et al. 2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

Self Cite

236

329

View full text Add to dashboard Cite

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 ⊆ N bywhere 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 inÕ(2 n log M ) time; the notationÕ suppresses polylogarithmic factors. Furthermore, using a standard embedding technique we can compute the subset convolution over the max-sum or min-sum semiring inÕ(2 n M ) time. To demonstrate the applicability of fast subset convolution, we present the firstÕ(2 k 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 theÕ(3 k n+2 k n 2 +nm) time bound of the classical DreyfusWagner algorithm. We also discuss extensions to recent O(2 n )-time algorithms for covering and partitioning problems (Björklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

show abstract

Section: Application To Specific Computational Problemsmentioning

confidence: 92%

Fourier meets möbius: fast subset convolution

Björklund

Husfeldt

Kaski

et al. 2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

Self Cite

236

329

View full text Add to dashboard Cite

show abstract

“…They also present a 2 n n O(1) time and space algorithm for counting k-colorings of a graph. Analogous results were also obtained by Koivisto [23].…”

Section: Introductionsupporting

confidence: 75%

Algorithms for Counting 2-Sat Solutions and Colorings with Applications

Fürer

Kasiviswanathan

Lecture Notes in Computer Science

View full text Add to dashboard Cite

An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-Cnf formula. The worst case running time of O(1.246 n ) for formulas with n variables improves on the previous bound of O(1.256 n ) by Dahllöf, Jonsson, and Wahlström. The algorithm uses only polynomial space. As a consequence we get an O(1.246 n ) time algorithm for counting maximum weighted independent sets.The above result when combined with a better partitioning technique for domains, leads to improved running times for counting the number of solutions of binary constraint satisfaction problems for all domain sizes. For large domain size d we approach O((0.601d) n ) improving the previous best bound of O((0.622d) n ). We further improve this bound for counting 3-colorings in a graph. The upper bound of O(1.770 n ) for graphs with n vertices improves on the previous bound of O(1.788 n ) by Angelsmark and Jonsson.

show abstract

“…Eppstein [7] reduced the bound to O(2.4151 n ) and Byskov [5] to O(2.4023 n ). In two breakthrough papers last year, Björklund & Husfeldt [3] and Koivisto [14] independently devised 2 n n O(1) algorithms based on a combination of inclusion-exclusion and dynamic programming.…”

Section: Introductionmentioning

confidence: 99%

“…#k-Coloring (and its generalization known as Chromatic Polynomial) are among the oldest counting problem. Recently Björklund & Husfeldt [3] and Koivisto [14] have shown that the chromatic polynomial of a graph can be computed in time 2 n n O (1) .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved Exact Algorithms for Counting 3- and 4-Colorings

Fomin

Gaspers

Saurabh

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We introduce a generic algorithmic technique and apply it on decision and counting versions of graph coloring. Our approach is based on the following idea: either a graph has nice (from the algorithmic point of view) properties which allow a simple recursive procedure to find the solution fast, or the pathwidth of the graph is small, which in turn can be used to find the solution by dynamic programming. By making use of this technique we obtain the fastest known exact algorithms -running in time O(1.7272 n ) for deciding if a graph is 4-colorable and -running in time O(1.6262 n ) and O(1.9464 n ) for counting the number of k-colorings for k = 3 and 4 respectively.

show abstract

An O*(2^n ) Algorithm for Graph Coloring and Other Partitioning Problems via Inclusion--Exclusion

Abstract: We use the principle of inclusion and exclusion, combined

Cited by 62 publications

References 25 publications

Fourier meets möbius: fast subset convolution

Fourier meets möbius: fast subset convolution

Algorithms for Counting 2-Sat Solutions and Colorings with Applications

Improved Exact Algorithms for Counting 3- and 4-Colorings

Contact Info

Product

Resources

About