Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable techniques for computing estimates with rigorous guarantees for discrete integration remains the holy grail. The key contribution of this work addresses this scalability challenge via an efficient reduction of discrete integration to model counting. The proposed reduction is achieved via a significant increase in the dimensionality that, contrary to conventional wisdom, leads to solving an instance of the relatively simpler problem of model counting. Building on the promising approach proposed by Chakraborty et al, our work overcomes the key weakness of their approach: a restriction to dyadic weights. We augment our proposed reduction, called DeWeight, with a state of the art efficient approximate model counter and perform detailed empirical analysis over benchmarks arising from neural network verification domains, an emerging application area of critical importance. DeWeight, to the best of our knowledge, is the first technique to compute estimates with provable guarantees for this class of benchmarks.

In this paper, we make a concrete progress towards answering discrete integration queries.

Discrete integration is a fundamental problem in computer science that concerns the computation of discrete sums over exponentially large sets. Despite intense interest from researchers for over three decades, the design of scalable techniques for computing estimates with rigorous guarantees for discrete integration remains the holy grail. The key contribution of this work addresses this scalability challenge via an efficient reduction of discrete integration to model counting. The proposed reduction is achieved via a significant increase in the dimensionality that, contrary to conventional wisdom, leads to solving an instance of the relatively simpler problem of model counting. Building on the promising approach proposed by Chakraborty et al [9], our work overcomes the key weakness of their approach: a restriction to dyadic weights. We augment our proposed reduction, called DeWeight, with a state of the art efficient approximate model counter and perform detailed empirical analysis over benchmarks arising from neural network verification domains, an emerging application area of critical importance. DeWeight, to the best of our knowledge, is the first technique to compute estimates with provable guarantees for this class of benchmarks.

The runtime performance of modern SAT solvers is deeply connected to the phase transition behavior of CNF formulas. While CNF solving has witnessed significant runtime improvement over the past two decades, the same does not hold for several other classes such as the conjunction of cardinality and XOR constraints, denoted as CARD-XOR formulas. The problem of determining the satisfiability of CARD-XOR formulas is a fundamental problem with a wide variety of applications ranging from discrete integration in the field of artificial intelligence to maximum likelihood decoding in coding theory. The runtime behavior of random CARD-XOR formulas is unexplored in prior work. In this paper, we present the first rigorous empirical study to characterize the runtime behavior of 1-CARD-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a non-linear tradeoff between CARD and XOR constraints.