## Studies in Solution Sampling

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Citations: | 4 - 0 self |

### BibTeX

@MISC{Gogate_studiesin,

author = {Vibhav Gogate and Rina Dechter},

title = {Studies in Solution Sampling},

year = {}

}

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### Abstract

We introduce novel algorithms for generating random solutions from a uniform distribution over the solutions of a boolean satisfiability problem. Our algorithms operate in two phases. In the first phase, we use a recently introduced SampleSearch scheme to generate biased samples while in the second phase we correct the bias by using either Sampling/Importance Resampling or the Metropolis-Hastings method. Unlike state-of-the-art algorithms, our algorithms guarantee convergence in the limit. Our empirical results demonstrate the superior performance of our new algorithms over several competing schemes.

### Citations

7042 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ...uniform distribution P(x) over the solutions in a product factored form: P(x = (x1,...,xn)) = ∏ n i=1 Pi(xi|x1,...,xi−1). Then, we use a standard Monte Carlo (MC) sampler (also called logic sampling (=-=Pearl, 1988-=-)) to sample along the ordering O = 〈X1,...,Xn〉 implied by the product form. Namely, at each step, given a partial assignment (x1,...,xi−1) to the previous i − 1 variables, we assign a value to variab... |

1216 |
Monte Carlo sampling methods using Markov chains and their applications
- Hastings
- 1970
(Show Context)
Citation Context ... limit the sampling distribution of the resulting techniques is the target uniform distribution over the solutions. SampleSearch-MH The main idea in the Metropolis-Hastings (MH) simulation algorithm (=-=Hastings, 1970-=-) is to generate a Markov Chain whose limiting or stationary distribution is equal to the target distribution P. A Markov chain consists of a sequence of states and a transition-rule T(y|x) for moving... |

565 | P.: Markov logic networks
- Richardson, Domingos
(Show Context)
Citation Context ...ions uniformly from hard combinatorial problems. Our schemes are quite general and can be applied to complex graphical models like Bayesian and Markov networks which have deterministic relationships (=-=Richardson and Domingos, 2006-=-; Fishelson and Geiger, 2003). However, in this paper, we focus on boolean satisfiability problems only. Our work is motivated by a wide application of solution sampling in fields such as verification... |

410 |
Simulation and the Monte Carlo Method
- Rubinstein
- 1981
(Show Context)
Citation Context ...rove that the samples generated by our new schemes converge in the limit to the required uniform distribution over the solutions. This convergence is important because, as implied by sampling theory (=-=Rubinstein, 1981-=-), it is expected to yield a decreasing sampling error (and thus more accurate estimates) as the number of samples (or time) is increased. In contrast, pure SampleSearch will converge to the wrong bac... |

76 | SATLIB: An Online Resource for Research on SAT
- Hoos, Stützle
- 2000
(Show Context)
Citation Context ...(c) circuit instances and (d) flat graph coloring instances. The first three benchmarks are available from Cachet (Sang et al., 2005) and the flat graph coloring benchmarks are available from Satlib (=-=Hoos and Stützle, 2000-=-). Note that we used SAT problems whose solutions can be counted in relatively small amount of time because to compute the KL distance we have to count solutions to n+1 SAT problems for each formula h... |

32 |
Comment on “The calculation of posterior distributions by data augmentation”, by M.Tanner and W.H.Wang
- Rubin
- 1987
(Show Context)
Citation Context ...om proposition 1 and 2 and the Metropolis-Hastings theory (Hastings, 1970). SampleSearch-SIR We now discuss our second algorithm which augments SampleSearch with Sampling/Importance Resampling (SIR) (=-=Rubin, 1987-=-) yielding the SampleSearch-SIR technique. Standard SIR (Rubin, 1987) aims at drawing random samples from a target distribution P(x) by using a given proposal distribution Q(x) which satisfies P(x) > ... |

29 | Optimizing exact genetic linkage computations
- Fishelson, Geiger
- 2004
(Show Context)
Citation Context ...atorial problems. Our schemes are quite general and can be applied to complex graphical models like Bayesian and Markov networks which have deterministic relationships (Richardson and Domingos, 2006; =-=Fishelson and Geiger, 2003-=-). However, in this paper, we focus on boolean satisfiability problems only. Our work is motivated by a wide application of solution sampling in fields such as verification and probabilistic reasoning... |

28 | Heuristics for fast exact model counting
- Sang, Beame, et al.
- 2005
(Show Context)
Citation Context ...) = |Sxi|/|S| where Sxi is the set of solutions that the assignment Xi = xi participates in and S is the set of all solutions. The number of solutions for the SAT problems were computed using Cachet (=-=Sang et al., 2005-=-). After running various sampling algorithms, we get a set of solution samples φ from which we compute the approximate marginal distribution as: Pa(Xi = xi) = φ(xi)/|φ| where φ(xi) is the number of so... |

18 | Generating random solutions for constraint satisfaction problems
- Dechter, Kask
- 2002
(Show Context)
Citation Context ...cus on boolean satisfiability problems only. Our work is motivated by a wide application of solution sampling in fields such as verification and probabilistic reasoning as elaborated in earlier work (=-=Dechter et al., 2002-=-; Wei et al., 2004; Gomes et al., 2007). The solution sampling task is closely related to the #PComplete problem of counting the number of solutions of a satisfiability problem. In fact, it is easy to... |

17 | Approximate counting by sampling the backtrack-free search space
- Gogate, Dechter
- 2007
(Show Context)
Citation Context ...mpling, (b) A proposal distribution Q and (c) Backtrack-free distribution of Q. (c) the distribution of samples generated from SampleSearch converges to the backtrack-free distribution defined below (=-=Gogate and Dechter, 2007-=-). Definition 2 (Backtrack-free distribution). Given a distribution Q(x) = ∏ n i=1 Qi(xi|x1,...,xi−1) , an ordering O = 〈x1,...,xn〉 and a cnf formula F, the backtrackfree distribution is QF (x) = ∏ n ... |

16 | Near-Uniform sampling of combinatorial spaces using XOR constraints
- Gomes, Sabharwal, et al.
- 2007
(Show Context)
Citation Context ...nly. Our work is motivated by a wide application of solution sampling in fields such as verification and probabilistic reasoning as elaborated in earlier work (Dechter et al., 2002; Wei et al., 2004; =-=Gomes et al., 2007-=-). The solution sampling task is closely related to the #PComplete problem of counting the number of solutions of a satisfiability problem. In fact, it is easy to show that if one can sample solutions... |

9 | A new algorithm of sampling CSP solutions uniformly at random
- Gogate, Dechter
- 2006
(Show Context)
Citation Context ...solutions uniformly. Since bucket elimination is time and space exponential in a graph parameter called the treewidth, it cannot be used when the treewidth is large. Therefore, in a subsequent paper (=-=Gogate and Dechter, 2006-=-), we proposed a search based sampling scheme called SampleSearch. SampleSearch is a randomized backtracking procedure whose value selection is guided by sampling from approximate (polynomial time) so... |

8 |
Minisat v1.13–a SAT solver with conflict-clause minimization
- Eén, Sörensson
- 2003
(Show Context)
Citation Context ...-bound used and therefore we report results for i-bound of 1. Since we can use any systematic SAT solver as an underlying search procedure within SampleSearch, we chose to use the minisat SAT solver (=-=Sorensson and Een, 2005-=-). We experimented with the following competing schemes (a) SampleSearch (b) SampleSearch-MH, (c) SampleSearch-SIR with replacement, (d) SampleSearchSIR without replacement and (e) SampleSAT. Followin... |

7 |
Improved sampling-importance resampling and reduced bias importance sampling. Scandin.journ. ofstat., theoryand apps,2003
- Skare
(Show Context)
Citation Context ...tistics literature to the basic SIR framework. In the following subsection, we consider one such improvement known as the Improved SIR framework. Improved SampleSearch-SIR Under certain restrictions (=-=Skare et al., 2003-=-) prove that the convergence of SIR is proportional to O(1/N). To speed up this convergence to O(1/N2 ), they propose the Improved SIR framework. For our purposes, Improved SIR only changes the weight... |

2 | Hung (2000). The multiple-try method - Liu, Liang, et al. |