Robert G. Jeroslow

Solving propositional satisfiability problems

We describe an algorithm for the satisfiability problem of prepositional logic, which is significantly more efficient for this problem than is a general mixed-integer programming code. Our algorithm is a list processor using a tree-search method, and is based on Lovelands form of the algorithm of Davis and Putnam.

The polynomial hierarchy and a simple model for competitive analysis

The multi-level linear programs of Candler, Norton and Townsley are a simple class of sequenced-move games, in which players are restricted in their moves only by common linear constraints, and each seeks to optimize a fixed linear criterion function in his/her own continuous variables and those of other players. All data of the game and earlier moves are known to a player when he/she is to move. The one-player case is just linear programming.We show that questions concerning only the value of these games exhibit complexity which goes up all levels of the polynomial hierarchy and appears to increase with the number of players.For three players, the games allow reduction of theΣ2 andΠ2 levels of the hierarchy. These levels essentially include computations done with branch-and-bound, in which one is given an oracle which can instantaneously solve NP-complete problems (e.g., integer linear programs). More generally, games with (p + 1) players allow reductions ofΣp andΠp in the hierarchy.An easy corollary of these results is that value questions for two-player (bi-level) games of this type is NP-hard.

Canonical Cuts on the Unit Hypercube

In this paper we study some properties of the n-dimensional unit hypercube K. We define a distance function on the set V of vertices of K, and use it to construct a class of hyperplanes parallel to the faces of K (canonical hyperplanes). We then establish some properties of these hyperplanes and of the associated canonical inequalities (cuts), and we show that adjacent canonical inequalities imply stronger canonical cuts. An arbitrary inequality is shown to imply a set of canonical cuts such that a vertex of K satisfies the arbitrary inequality if and only if it satisfies the set of implied canonical cuts (Theorem 2). As a consequence of this, we show that every bounded integer program can be stated as a set covering problem (Corollary 2.1). The above results (with the exception of Corollary 2.1, which was added later) were first stated in [1], and a brief note on them was published in [2]. For background material and related concepts the reader is referred to [3].

Some results and experiments in programming techniques for propositional logic

Abstract We provide some results which relate mathematical programming techniques for treating propositional logic to theorem-proving techniques for this logic. We show that the branch-and-bound method, applied to the usual linear representation of clauses, is very similar to an implementation of Lovelands (generally stronger) form of the algorithm of Davis and Putnam. However, branch-and-bound does not include the fixing of monotone variables, while it does have the incumbent-finding feature which is lacking in the Davis-Putnam procedure. We also provide a linear-time algorithm for achieving a clausal form equivalent to a given proposition, in some additional atomic letters. Experimental results are included.

The value function of an integer program

We consider integer programs in which the objective function and constraint matrix are fixed while the right-hand side varies. The value function gives, for each feasible right-hand side, the criterion value of the optimal solution. We provide a precise characterization of the closed-form expression for any value function.The class of Gomory functions consists of those functions constructed from linear functions by taking maximums, sums, non-negative multiples, and ceiling (i.e., next highest integer) operations.The class of Gomory functions is identified with the class of all possible value functions by the following results: (1) for any Gomory functiong, there is an integer program which is feasible for all integer vectorsv and hasg as value function; (2) for any integer program, there is a Gomory functiong which is the value function for that program (for all feasible right-hand sides); (3) for any integer program there is a Gomory functionf such thatf(v)≤0 if and only ifv is a feasible right-hand side. Applications of (1)–(3) are also given.

Strengthening cuts for mixed integer programs

Abstract We give a method for strengthening cutting planes for pure and mixed integer programs. The method improves the coefficients of the integer-constrained variables, while leaving unchanged those of the continuous variables. We first state the general principle on which the method is based; then apply it to the class of cuts that can be obtained from disjunctive constraints. Finally, we give simple procedures for calculating the improved coefficients of cats in this class, and illustrate them on a numerical example.

There Cannot be any Algorithm for Integer Programming with Quadratic Constraints

This paper studies a class of integer programming problems in which squares of variables may occur in the constraints, and shows that no computing device can be programmed to compute the optimum criterion value for all problems in this class.

Representability in mixed integer programming, I: characterization results

Abstract We define the concept of a representation of a set of either linear constraints in bounded integers, or convex constraints in bounded integers. A regularity condition plays a crucial role in the convex case. Then we characterize the representable sets (Theorem 2.1) and provide several examples of our representations. A consequence of our characterization is that the only representable sets are those from ‘either/or’ constraints. This latter case can be treated by generalizations of techniques from the disjunctive methods of cutting-plane theory (e.g. [2] and [30]). The representations given here are intended for use as part of the constraints of a larger optimization problem, where they often can serve to tighten the (linear or convex) relaxation. The study of representations was initiated by Meyer and in the linear case we continue the development in [35].

Cutting-Plane Theory: Disjunctive Methods

This paper is a survey, with new results, of the disjunctive methods of cutting-plane theory, which were devised by Balas, Glover, Owen, Young, and other researchers, over the past half decade. The basic disjunctive cut principle is derived, its interrelations with the other cut-producing procedures are discussed, and applications of it are given. Many theorems from the literature are concisely proven, and a fairly complete bibliography is provided. In addition, several new results are presented, and finitely convergent disjunctive cutting-plane algorithms are given for a wide class of programs.

Computation-oriented reductions of predicate to propositional logic

Abstract We give a sequential reduction technique for transforming a formula of the pure predicate calculus to a sequence of ‘increasingly accurate’ propositional logic ‘approximations’ to the predicate formula. At each stage of approximation, one of the following three cases occurs: (1) the approximation shows the predicate formula to be satisfiable; or (2) the approximation shows the predicate formula to be unsatisfiable; or (3) neither (1) nor (2) occurs, and a ‘more accurate’ refinement of the current propositional ‘approximation’ is obtained. The overall reduction technique is finite for the ∀- , ∃- , and ∃ ∀ satisfiability fragments of pure logic. In fact, it achieves the proven complexity bounds of Lewis [38] for these fragments, where nondeterminism is replaced by exponentiation. Since discrete programming and mixed integer programming can be used to treat propositional logic, this sequential reduction technique opens further possibilities for applications of mathematical programming to logic-based methods of decision support. In this respect, there is particular interest in the combination of logic with the nonlogical side constraints typical in mathematical programming, which are not efficiently handled by logic alone (e.g., material balances, capacity restrictions, demand requirements, etc.).