Satisfiability and Computing van der Waerden Numbers

Abstract

Abstract. In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) in question. We show that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective -competitive with specialized approaches. By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers. We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers. IntroductionIn this paper we discuss how the areas of propositional satisfiability and combinatorics can help advance each other. On one hand, we show that recent dramatic improvements in the efficiency of SAT solvers and their extensions make it possible to obtain new results in combinatorics simply by encoding problems as propositional theories, and then computing their models (or deciding that none exist) using off-the-shelf general-purpose SAT solvers. On the other hand, we argue that combinatorics is a rich source of structured, parameterized families of hard propositional theories, and can provide useful sets of benchmarks for developing and testing new generations of SAT solvers.In our paper we focus on the problem of computing van der Waerden numbers. The celebrated van der Waerden theorem [20] asserts that for every positive integers k and l there is a positive integer m such that every partition of {1, . . . , m} into k blocks (parts) has at least one block with an arithmetic progression of length l. The problem is to find the least such number m. This number is called the van der Waerden number W (k, l). Exact values of W (k, l) are known only for five pairs (k, l). For other combinations of k and l there are some general lower and upper bounds but they are very coarse and do not give any good idea about the actual value of W (k, l). In the paper we show that SAT solvers such as POSIT [6], and SATO [21], as well as recently developed local-search solver walkaspps [13], designed to compute models for propositional theories extended by cardinality atoms [4], can improve lower bounds for van der Waerden numbers for several combinations of parameters k and l.Theories that arise in these investigations are determined by the two parameters k and l. Therefore, they show a substantial degree of structure and similarity. Moreover, as k and l grow, these theories ...

Bounds on some van der Waerden numbers

“…The proof is essentially the same as the proof given by Graham [7] of an upper bound for w 1 (k, 3). For completeness, we include the proof here.…”

“…A list of other known exact values of w(k, m; 2) appears in [15]. Improved lower bounds on several specific values of w(k, k; s) are given in [3] and [10].…”

On Computation of Exact van der Waerden Numbers

“…For both these numbers, the encoding in [1,8] takes more than a couple of months to prove the corresponding instances unsatisfiable.…”

WSAT(CC) — A Fast Local-Search ASP Solver

“…In [4], we showed that WSAT (CC ) is often much faster than a local-search SAT solver WSAT and has, in general, a higher success rate (likelihood that it will find a model if an input theory has one). In [1], we used WSAT (CC ) to compute several new lower bounds for van der Waerden numbers. Here, we will discuss our recent comparisons of WSAT (CC ) with WSAT (OIP ) [7], a solver for propositional theories extended with pseudo-boolean constraints (for which we developed utilities allowing it to accept PL cc theories).…”