## A Tighter Bound for Counting Max-Weight Solutions to 2SAT Instances

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Citations: | 7 - 0 self |

### BibTeX

@MISC{Wahlström_atighter,

author = {Magnus Wahlström},

title = {A Tighter Bound for Counting Max-Weight Solutions to 2SAT Instances},

year = {}

}

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### Abstract

We give an algorithm for counting the number of max-weight solutions to a 2SAT formula, and improve the bound on its running time to O (1.2377 n). The main source of the improvement is a refinement of the method of analysis, where we extend the concept of compound (piecewise linear) measures to multivariate measures, also allowing the optimal parameters for the measure to be found automatically. This method extension should be of independent interest.

### Citations

506 |
The Complexity of Computing the Permanent
- Valiant
- 1979
(Show Context)
Citation Context ...omplexity point of view, the problem class #P of problems where you want to know the number of solutions to some problem in NP is a very difficult one. The class was proposed by Valiant in the 1970’s =-=[14]-=-, and it was later proved that the so-called polynomial hierarchy is contained in P #P [12] (i.e. that a polynomial-time algorithm for any #P-complete problem would allow us to solve any problem in th... |

421 |
The Complexity of Enumeration and Reliability Problems
- Valiant
- 1979
(Show Context)
Citation Context ...hat are in P. The problem considered in this paper, #2SAT, is an example of the latter: the “decision variant” 2SAT is a well-known polynomial problem, while the counting version #2SAT is #P-complete =-=[9, 13]-=-. However, despite the apparent difficulty of the class, individual #P-complete problems can be solved in reasonable exponential time; for instance, the bound O ∗ (1.2377 n ) 1 for #2SAT is significan... |

191 |
PP is as Hard as the Polynomial-Time Hierarchy
- Toda
- 1991
(Show Context)
Citation Context ... of solutions to some problem in NP is a very difficult one. The class was proposed by Valiant in the 1970’s [14], and it was later proved that the so-called polynomial hierarchy is contained in P #P =-=[12]-=- (i.e. that a polynomial-time algorithm for any #P-complete problem would allow us to solve any problem in the polynomial hierarchy in polynomial time; in fact, a single query to the algorithm would s... |

130 |
The design and analysis of algorithms
- Kozen
- 1992
(Show Context)
Citation Context ...hat are in P. The problem considered in this paper, #2SAT, is an example of the latter: the “decision variant” 2SAT is a well-known polynomial problem, while the counting version #2SAT is #P-complete =-=[9, 13]-=-. However, despite the apparent difficulty of the class, individual #P-complete problems can be solved in reasonable exponential time; for instance, the bound O ∗ (1.2377 n ) 1 for #2SAT is significan... |

39 | Improved upper bounds for 3-sat
- Iwama, Tamaki
- 2004
(Show Context)
Citation Context ... bound O ∗ (1.2377 n ) 1 for #2SAT is significantly faster than any bound for solving 3SAT (for which the best bounds are a probabilistic algorithm with a bound of O ∗ (1.3238 n ) by Iwama and Tamaki =-=[8]-=-, and a deterministic algorithm with a bound of O ∗ (1.473 n ) by Brueggemann and Kern [1]). One of the first algorithms for a counting problem came in the early 1960’s with Ryser’s [11] O(n 2 2 n ) t... |

29 | Counting the number of solutions for instances of satisfiability - Dubois - 1991 |

28 | Some new techniques in design and analysis of exact (exponential) algorithms
- Fomin, Grandoni, et al.
- 2005
(Show Context)
Citation Context ...a non-negative complexity value to any possible instance F of some problem. They are used for estimating upper bounds on the running times of branching algorithms; see e.g. the survey by Fomin et al. =-=[6]-=-. For a branching algorithm and a complexity measure µ(F ), the branching number of a particular branching from an instance F to subinstances F1, . . . , Fk, with µ(F ) − µ(Fi) = δi, is τ(δ1, . . . , ... |

26 | Quasiconvex analysis of backtracking algorithms
- Eppstein
- 2003
(Show Context)
Citation Context ...provement of the analysis through compound measures, this time introducing multi-variate compound measures, which are a combination of compound measures with the multi-variate recurrences of Eppstein =-=[5]-=-. Analysis through compound measures allows us to model that the behaviour of the algorithm varies depending on certain parameters on the instance, in this case the average degree, i.e. that the behav... |

18 |
An improved deterministic local search algorithm for 3-SAT
- Brueggemann, Kern
- 2004
(Show Context)
Citation Context ... (for which the best bounds are a probabilistic algorithm with a bound of O ∗ (1.3238 n ) by Iwama and Tamaki [8], and a deterministic algorithm with a bound of O ∗ (1.473 n ) by Brueggemann and Kern =-=[1]-=-). One of the first algorithms for a counting problem came in the early 1960’s with Ryser’s [11] O(n 2 2 n ) time algorithm for counting the number of perfect matchings in a bipartite graph (also know... |

18 | Algorithms for counting 2-SAT solutions and colorings with applications
- Fürer, Kasiviswanathan
- 2005
(Show Context)
Citation Context ...ious work on the #2SAT problem with better bounds than O ∗ (2 n ) includes results by Dubois [4], Zhang [17], Littman et al. [10], Dahllöf, Jonsson and Wahlström [2, 3], and Fürer and Kasiviswanathan =-=[7]-=-. In terms 1 O ∗ (·), Θ ∗ (·), etc, signify that polynomial factors are ignoredsof the actual bounds, the bound O ∗ (1.3247 n ) appeared in [2]; later, a complete rewrite of the algorithm for the jour... |

13 | Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems
- Wahlström
- 2007
(Show Context)
Citation Context ...ect. 5 providing the analysis of upper bounds for maximum degree four, and finally Sect. 6 containing the general upper bound. Major portions of this paper appeared as Chapt. 7 in the author’s thesis =-=[16]-=-, but the material has not been published in any referreed publication. Some proofs have been omitted due to lack of space; these can usually be found in [16]. 2 Preliminaries A variable, in this pape... |

11 |
of models and satisfiability of sets of clauses
- Zhang
- 1996
(Show Context)
Citation Context ...ect matchings in a bipartite graph (also known as computing the permanent of a 0/1 matrix). Previous work on the #2SAT problem with better bounds than O ∗ (2 n ) includes results by Dubois [4], Zhang =-=[17]-=-, Littman et al. [10], Dahllöf, Jonsson and Wahlström [2, 3], and Fürer and Kasiviswanathan [7]. In terms 1 O ∗ (·), Θ ∗ (·), etc, signify that polynomial factors are ignoredsof the actual bounds, the... |

9 |
Wahlström M., Counting models for 2SAT
- Dahllöf, Jonsson
- 2005
(Show Context)
Citation Context ...g the permanent of a 0/1 matrix). Previous work on the #2SAT problem with better bounds than O ∗ (2 n ) includes results by Dubois [4], Zhang [17], Littman et al. [10], Dahllöf, Jonsson and Wahlström =-=[2, 3]-=-, and Fürer and Kasiviswanathan [7]. In terms 1 O ∗ (·), Θ ∗ (·), etc, signify that polynomial factors are ignoredsof the actual bounds, the bound O ∗ (1.3247 n ) appeared in [2]; later, a complete re... |

5 |
On the complexity of counting satisfying assignments
- Littman, Pitassi, et al.
- 2001
(Show Context)
Citation Context ...partite graph (also known as computing the permanent of a 0/1 matrix). Previous work on the #2SAT problem with better bounds than O ∗ (2 n ) includes results by Dubois [4], Zhang [17], Littman et al. =-=[10]-=-, Dahllöf, Jonsson and Wahlström [2, 3], and Fürer and Kasiviswanathan [7]. In terms 1 O ∗ (·), Θ ∗ (·), etc, signify that polynomial factors are ignoredsof the actual bounds, the bound O ∗ (1.3247 n ... |

4 | Counting satisfying assignments in 2-SAT and 3-SAT
- Dahllöf, Jonsson, et al.
(Show Context)
Citation Context ...g the permanent of a 0/1 matrix). Previous work on the #2SAT problem with better bounds than O ∗ (2 n ) includes results by Dubois [4], Zhang [17], Littman et al. [10], Dahllöf, Jonsson and Wahlström =-=[2, 3]-=-, and Fürer and Kasiviswanathan [7]. In terms 1 O ∗ (·), Θ ∗ (·), etc, signify that polynomial factors are ignoredsof the actual bounds, the bound O ∗ (1.3247 n ) appeared in [2]; later, a complete re... |

4 |
An algorithm for the SAT problem for formulae of linear length
- Wahlstrom
- 2005
(Show Context)
Citation Context ...f the algorithm is non-uniform. Apart from earlier #2SATw publications, this type of analysis has been applied to the problem of finding a solution to SAT instances F with a bounded ℓ(F )/n(F ) value =-=[15]-=-. Combining the method with Eppstein’s method for solving multi-variate recurrence improves the quality of the bound, and allows us to automate the bound calculations. The paper is structured into Sec... |