A Compact and Efficient SAT-Encoding of Finite Domain CSP

Extended Abstract A (finite) Constraint Satisfaction Problem (CSP) is a combinatorial problem to find an assignment which satisfies all given constraints over finite domains. A SAT-based CSP solver is a program which solves a CSP by encoding it to SAT and searching solutions by SAT solvers. Remarkable improvements in the efficiency of SAT solvers make SAT-based CSP solvers applicable for solving hard and practical problems. A number of SAT encoding methods have been therefore proposed: direct encoding, support encoding, log encoding, log-support encoding, and order encoding. Among them, order encoding [4] has showed a good performance for a wide variety of problems, including Open-Shop Scheduling problems, two-dimensional strip packing problems, and test case generation. Its effectiveness has also been shown by the fact that a SAT-based CSP solver Sugar 3 became a winner in several categories of the 2008 and 2009 International CSP Solver Competitions. However, in the order encoding, the size of SAT-encoded instances becomes huge when the domain size of the original CSP is large. On the other hand, the log encoding [3, 1] uses a bit-wise representation for integer variables. The size of SAT-encoded instances is therefore compact (linear to log d), but its performance is slow in general because it requires many inference steps to " ripple " carries. In this paper, we propose a new encoding, named compact order encoding, aiming to be compact and efficient. The basic idea of the compact order encoding is the use of a numeric system of base B ≥ 2. That is, each integer variable x is represented by a summation ∑ m−1 i=0 B i x i where m = log B d and 0 ≤ x i < B for all x i , and each x i is encoded by the order encoding. Each ternary constraints of addition and multiplication can be encoded into at most O(B 2 log B d) and O(B 3 log B d + B 2 log 2 B d) clauses respectively which are much less than O(d 2) clauses of the order encoding. The compact order encoding can generate much efficient SAT instance than the log encoding in general because it requires fewer carry propagations. Please note that the compact order encoding with base B = 2 is equivalent to the log encoding, and the one with base B ≥ d is equivalent to the order encoding.