Chapter 26. Approximate Model Counting

Model counting, or counting solutions of a set of constraints, is a fundamental problem in Computer Science with diverse applications. Since exact counting is computationally hard (#P complete), approximate counting techniques have received much attention over the past few decades. In this chapter, we focus on counting models of propositional formulas, and discuss in detail universal-hashing based approximate counting, which has emerged as the predominant paradigm for state-of-the-art approximate model counters. These counters are randomized algorithms that exploit properties of universal hash functions to provide rigorous approximation guarantees, while piggybacking on impressive advances in propositional satisfiability solving to scale up to problem instances with a million variables. We elaborate on various choices in designing such approximate counters and the implications of these choices. We also discuss variants of approximate model counting, such as DNF counting and weighted counting.

From Support Propagation to Belief Propagation in Constraint Programming (Extended Abstract)

The distinctive driving force of constraint programming (CP) to solve combinatorial problems has been a privileged access to problem structure through the high-level models it uses. We investigate a richer propagation medium for CP made possible by recent work on counting solutions inside constraints. Beliefs about individual variable-value assignments are exchanged between contraints and iteratively adjusted. Its advantage over standard belief propagation is that the higher-level models do not tend to create as many cycles, which are known to be problematic for convergence. We find that it significantly improves search guidance.

Revisiting Counting Solutions for the Global Cardinality Constraint

Counting solutions for a combinatorial problem has been identified as an important concern within the Artificial Intelligence field. It is indeed very helpful when exploring the structure of the solution space. In this context, this paper revisits the computation process to count solutions for the global cardinality constraint in the context of counting-based search. It first highlights an error and then presents a way to correct the upper bound on the number of solutions for this constraint.

Remarks on Chebyshev polynomials, Fibonacci polynomials and Kauffman bracket skein modules

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.