Linear Programming Formulation of the Boolean Satisfiability Problem

Boolean satisfiability (SAT) is of central importance in many areas of Operations Research and Computer Science. In its general (conjunctive normal) form, the problem can be stated as follows. There is a number of Boolean (binary) variables generically called “literals,” and a number of covertype constraints implicitly defined over these variables, called “clauses.” Pairs of the literals may be negations of each other. For example, if x1 is a (positive) literal, the corresponding “negative” literal is not-x1 (or qx1). A clause consists of a subset of the literals, and evaluates to TRUE if one or more of the literals in the subset are set TRUE. A conjunction of clauses is referred to as a “propositional” or “Boolean” formula. The problem is to find a “truth assignment” to the literals so that a given Boolean formula evaluates to TRUE, or to determine that no such assignment exists. Practical applications of this problem abound in Operations Research and Computer Science (see [4] for examples). A version of the problem that is often used in theoretical developments is the “1-in-3 SAT” (or the “exactly-1 3SAT”). In 1-in-3 SAT, each clause has exactly 3 literals, and a clause evaluates to TRUE iff exactly one of its literals is set (i.e., assigned a value of) TRUE. The reason for using 1-in-3 SAT in theoretical developments is that the general SAT is polynomially transformable to 1-in-3 SAT ([8]). Also, because SAT was shown to be NP-complete (in fact, it was the first problem to be so [2]), the focus of research has been on the development of efficient enumeration schemes and heuristics (see [6] for an extensive survey). In this paper, we develop a linear programming (LP) model of a generalized version of 1-in-3 SAT where clauses are allowed to have arbitrary numbers of literals, respectively. We refer to this problem as “exactly-1 SAT.” We use a bipartite network flow (BNF)-based model we develop, and a path-based modeling approach similar to that used in [3] to formulate this problem as a linear program. The approach is illustrated with a numerical example. The plan of the paper is as follows. The BNF-based model is discussed in section 2. The path-based formulation is discussed in section 3. The overall LP model is discussed in section 4. Conclusions are discussed in section 5.