Algorithms for Counting 2-Sat Solutions and Colorings with Applications

Fürer, Martin; Kasiviswanathan, Shiva Prasad

doi:10.1007/978-3-540-72870-2_5

Cited by 37 publications

(39 citation statements)

References 28 publications

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“…This improves on the O(1.770 n ) time algorithm presented in [20]. We can also use this to obtain faster algorithms for counting k-colorings as done in [1].…”

Section: Counting 3-colorings (#3-coloring)mentioning

confidence: 87%

“…Notable contributions include papers by Williams [38] and Fürer and Kashiviswanathan [20]. The current fastest algorithm for #3-Coloring has running time O(1.770 n ) [20]. Here we improve this algorithm with our technique of combining branching and treewidth and give an O(1.6308 n ) time algorithm for the problem.…”

Section: Introductionmentioning

confidence: 99%

“…In recent years, the algorithms for (d, 2)-CSP have been significantly improved. Notable contributions include papers by Williams [38] and Fürer and Kashiviswanathan [20]. The current fastest algorithm for #3-Coloring has running time O(1.770 n ) [20].…”

Section: Introductionmentioning

confidence: 99%

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On Two Techniques of Combining Branching and Treewidth

et al. 2007

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Abstract. Branch & Reduce and dynamic programming on graphs of bounded treewidth are among the most common and powerful techniques used in the design of moderately exponential time exact algorithms for NP hard problems. In this paper we discuss the efficiency of simple algorithms based on combinations of these techniques. The idea behind these algorithms is very natural: If a parameter like the treewidth of a graph is small, algorithms based on dynamic programming perform well. On the other side, if the treewidth is large, then there must be vertices of high degree in the graph, which is good for branching algorithms. We give several examples of possible combinations of branching and programming which provide the fastest known algorithms for a number of NP hard problems. All our algorithms require non-trivial balancing of these two techniques.In the first approach the algorithm either performs fast branching, or if there is an obstacle for fast branching, this obstacle is used for the construction of a path decomposition of small width for the original graph. Using this approach we give the fastest known algorithms for Minimum Maximal Matching and for counting all 3-colorings of a graph.In the second approach the branching occurs until the algorithm reaches a subproblem with a small number of edges (and here the right choice of the size of subproblems is crucial) and then dynamic programming is applied on these subproblems of small width. We exemplify this approach by giving the fastest known algorithm to count all minimum weighted dominating sets of a graph.We also discuss how similar techniques can be used to design faster parameterized algorithms.

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“…This improves on the O(1.770 n ) time algorithm presented in [20]. We can also use this to obtain faster algorithms for counting k-colorings as done in [1].…”

Section: Counting 3-colorings (#3-coloring)mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

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On Two Techniques of Combining Branching and Treewidth

et al. 2007

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“…Evaluating the polynomial χ(G, λ) for integers λ ≥ 3 is Pcomplete, [Val79]. For the best known algorithms, see [FK03]. Even if restricted to planar graphs, counting the number of 3-colorings or the number of acyclic orientations, i.e.…”

Section: Complexity Of Computing the Graph Polynomialsmentioning

confidence: 99%

Computing Graph Polynomials on Graphs of Bounded Clique-Width

Makowsky

Rotics

Averbouch

et al. 2006

Graph-Theoretic Concepts in Computer Science

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Abstract. We discuss the complexity of computing various graph polynomials of graphs of fixed clique-width. We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n f (k) ), where f (k) ≤ 3 for the inerlace polynomial, f (k) ≤ 2k + 1 for the matching polynomial and f (k) ≤ 3 · 2 k+2 for the chromatic polynomial.

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“…Recently also counting versions of weighted SAT have been considered: #XSAT for variable weighted CNF formulas can be solved in time O(n 2 · C +2 0.40567·n ) as shown in [14,15]. Fürer et al [5] provided an algorithm for counting all maximum weight solutions of SAT for variable weighted 2-CNF formulas. Clearly, only counting models cannot provide a solution of the underlying optimization problem, as no solutions are generated explicitly.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Variable-Weighted 2-SAT and Dual Problems

Porschen

Speckenmeyer

Theory and Applications of Satisfiability Testing – SAT 2007

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Abstract. In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems.

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Algorithms for Counting 2-Sat Solutions and Colorings with Applications

Cited by 37 publications

References 28 publications

On Two Techniques of Combining Branching and Treewidth

On Two Techniques of Combining Branching and Treewidth

Computing Graph Polynomials on Graphs of Bounded Clique-Width

Algorithms for Variable-Weighted 2-SAT and Dual Problems

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