Abstract. A promising approach for model counting was recently introduced, which in theory requires the use of large random xor or parity constraints to obtain near-exact counts of solutions to Boolean formulas. In practice, however, short xor constraints are preferred as they allow better constraint propagation in SAT solvers. We narrow this gap between theory and practice by presenting experimental evidence that for structured problem domains, very short xor constraints can lead to probabilistic variance as low as large xor constraints, and thus provide the same correctness guarantees. We initiate an understanding of this phenomenon by relating it to structural properties of synthetic instances. 1

Abstract. In this paper, we describe a new algorithm that approximately solves the problem of sampling solutions from a uniform distribution over the solutions of a constraint network. Our new algorithm improves upon the Sampling/Importance Resampling (SIR) component of our previous scheme of SampleSearch-SIR by taking advantage of the decomposition implied by the network’s AND/OR search space. We describe how our new scheme can be modified to approximately count and lower bound the number of solutions of a constraint network. We demonstrate both theoretically and empirically that on networks which have a favorable decomposition, our new algorithm yields far better performance than competing approaches. 1

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.

Abstract. We introduce a new technique of SampleSearch-SIR to generate random solutions of a Boolean satisfiability problem from a uniform distribution over the solutions. Our technique operates in two phases. In the first phase, it uses a recently proposed SampleSearch scheme to generate approximately random solutions from the satisfiability problem and then in the second phase it uses the Sampling Importance Resampling (SIR) principle to reduce the approximation error introduced by SampleSearch. The use of SIR guarantees convergence (in the limit) that none of the current state-of-the-art schemes have. Our empirical results demonstrate the superior performance and better convergence of SampleSearch-SIR as compared to state-of-the-art schemes. 1

for inspiring me to consider various fields of research and providing feedback on my projects and papers. I also want to thank my defense committee for their comments and insights: Professor John Hayes, Professor Karem Sakallah, and Professor Dennis Sylvester. I would like to thank Professor David Kieras for enhancing my knowledge and appreciation for computer programming and providing invaluable advice. Over the years, I have been fortunate to know and work with several wonderful students. I have collaborated extensively with Kai-hui Chang and Smita Krishnaswamy and have enjoyed numerous research discussions with them and have benefited from their insights. I would like to thank Ian Kountanis and Zaher Andraus for our many fun discussions on parallel SAT. I also appreciate the time spent collaborating with Kypros Constantinides and Jason Blome. Although I have not formally collaborated with Ilya Wagner, I have enjoyed numerous discussions with him during my doctoral studies. I also thank my office mates Jarrod Roy, Jin Hu, and Hector Garcia. Without my family and friends I would never have come this far. I would like to thank Geoff Blake and Smita Krishnaswamy, who have been both good friends and colleagues

Abstract. This document is an addendum to [1]. We complement Section 4 of [1] in two ways. First, a detailed analysis of the randomized algorithms is provided. Second, an explanation of the behavior of the quality functions of pguide and pbcpguide 100 is proposed. The reader should be familiar with the content of [1] up to and including Section 4. 1 Analyzing Randomized Algorithms This section complements the analysis presented in Section 4 of [1] by analyzing the behavior of three randomized algorithms and comparing them to prand. dpll-based sampling invokes the SAT solver k times to generate k models for the same input formula. The first assignment to a variable is random for each invocation of the SAT solver. dpll-based sampling was mentioned in [2], but we did not find any reference to work introducing it. xor-sample [3] invokes the SAT solver at least k times to generate k models. For each invocation, the initial formula is augmented with random XOR constraints, where an XOR constraint includes variables and, optionally, the

Computing the market maker price of a security in a combinatorial prediction market is #P-hard. We devise a fully polynomial randomized approximation scheme (FPRAS) that computes the price of any security in disjunctive normal form (DNF) within an ɛ multiplicative error factor in time polynomial in 1/ɛ and the size of the input, with high probability and under reasonable assumptions. Our algorithm is a Monte-Carlo technique based on importance sampling. The algorithm can also approximately price securities represented in conjunctive normal form (CNF) with additive error bounds. To illustrate the applicability of our algorithm, we show that many securities in Yahoo!’s popular combinatorial prediction market game called Predictalot can be represented by DNF formulas of polynomial size.