New Plain-Exponential Time Classes for Graph Homomorphism

Wahlström, Magnus

doi:10.1007/s00224-010-9261-z

Cited by 16 publications

(26 citation statements)

References 23 publications

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“…Recently, Wahlström [49] proved that if the clique-width of a graph F is at most c, then hom(F, G) can be computed in time ((2c + 1) n F + 2 cn ) · n O(1) , where n F and n is the number of vertices in F and G, correspondingly. By the results of this paper, it implies that sub(F, G) can be computed in time 2 O(n) , when the clique-width of F is constant.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Counting Subgraphs via Homomorphisms

Amini

Fomin

Saurabh

2009

Automata, Languages and Programming

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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 subgraphs excluding a fixed graph M as a minor can be done in 2 O(n) time.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Counting Subgraphs via Homomorphisms

Amini

Fomin

Saurabh

2009

Automata, Languages and Programming

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“…And as we observed already, for H-coloring, the brute-force algorithm solving H-coloring runs in time 2 O(n log h) . In spite of all the similarities between graph coloring and homomorphism, no substantially faster algorithm was known and it was an open question in the area of exact algorithms if there is a single-exponential algorithm solving H-coloring in time 2 O(n+h) [Fomin et al 2007;Rzażewski 2014;Wahlström 2010;2011], see also [Fomin and Kratsch 2010, Chapter 12].…”

Section: Graph Homomorphismmentioning

confidence: 99%

Tight Lower Bounds on Graph Embedding Problems

Cygan

Fomin

Golovnev³

et al. 2017

J. ACM

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We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph G to graph H cannot be done in timeWe also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of |V (H)| o(|V (H)|) -time algorithm deciding if graph G is a subgraph of H. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

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“…In fact, it is even unknown whether graphs of clique-width at most 4 can be recognized in polynomial time [Corneil et al 2012]. One can decide in exponential time (2k + 1) n n O(1) whether the clique-width of a graph with n vertices is at most k [Wahlström 2011]. There are approximation algorithms with an exponential error that, for fixed k, compute f (k)-expressions for graphs of clique-width at most k in polynomial time (where f (k) = (2 3k+2 − 1) by Oum and Seymour [2006], and f (k) = 8 k − 1 by Oum [2008]).…”

Section: Introductionmentioning

confidence: 99%

A SAT Approach to Clique-Width

Heule

Szeider

2015

ACM Trans. Comput. Logic

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Clique-width is a graph invariant that has been widely studied in combinatorics and computational logic. Computing the clique-width of a graph is an intricate problem, because the exact clique-width is not known even for very small graphs. We present a new method for computing clique-width via an encoding to propositional satisfiability (SAT), which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SATbased method is the first to discover the exact clique-width of various small graphs, including famous named graphs from the literature as well as random graphs of various density. With our method, we determined the smallest graphs that require a small predescribed clique-width. We further show how our method can be modified to compute the linear clique-width of graphs, a variant of clique-width that has recently received considerable attention. In an appendix, we provide certificates for tight upper bounds for the clique-width and linear clique-width of famous named graphs.

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New Plain-Exponential Time Classes for Graph Homomorphism

Abstract: Abstract. A homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f : V (G)

Cited by 16 publications

References 23 publications

Counting Subgraphs via Homomorphisms

Counting Subgraphs via Homomorphisms

Tight Lower Bounds on Graph Embedding Problems

A SAT Approach to Clique-Width

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