We apply the zero-one integer programming algorithm described in Karmarkar [12] and Karmarkar, Resende and Ramakrishnan [13] to solve randomly generated instances of the satisfiability problem (SAT). The interior point algorithm is briefly reviewed and shown to be easily adapted to solve large instances of SAT. Hundreds of instances of SAT (having from 100 to 1000 variables and 100 to 32,000 clauses) are randomly generated and solved. For comparison, we attempt to solve the problems via linear programming relaxation with MINOS.

In this paper we describe an interior point mathematical programming approach to inductive inference. We list several versions of this problem and study in detail the formulation based on hidden Boolean logic. We consider the problem of identifying a hidden Boolean function F : f0; 1g n ! f0; 1g using outputs obtained by applying a limited number of random inputs to the hidden function. Given this input-output sample, we give a method to synthesize a Boolean function that describes the sample. We pose the Boolean Function Synthesis Problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. A similar integer programming implementation has been used in a previous study to solve randomly generated instances of the Satisfiability Problem. In this paper we introduce a new variant of this algorithm, where the Riemannian metric used...

We present an approximation algorithm for solving large 0-1 integer programming problems where A is 0-1 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems ...

Cutting plane methods require the solution of a sequence of linear programs, where the solution to one provides a warm start to the next. A cutting plane algorithm for solving the linear ordering problem is described. This algorithm uses the primal-dual interior point method to solve the linear programming relaxations. A point which isagoodwarm start for a simplex-based cutting plane algorithm is generally not a good starting point for an interior point method. Techniques used to improve the warm start include attempting to identify cutting planes early and storing an old feasible point, which is used to help recenter when cutting planes are added. Computational results are described for some real-world problems; the algorithm appears to be competitive with a simplex-based cutting plane algorithm.

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.

Given a Boolean satisfiability (Sat) problem whose variables have non-negative weights, the minimum-cost satisfiability (MinCostSat) problem finds a satisfying truth assignment that minimizes a weighted sum of the truth values of the variables. Many NP-optimization problems are either special cases of MinCostSat or can be transformed into MinCostSat efficiently. However, in the past, these problems have been largely considered in isolation. In this dissertation, we (1) classify existing Min-CostSat problems, (2) study factors affecting the performance of MinCostSat solvers, (3) propose algorithms for MinCostSat problems, and (4) implement and validate the performance of state-of-the-art solvers for special cases of MinCostSat, including set and binate covering, Max-Sat, and group-partial Max-Sat. We categorize MinCostSat problems as either native or non-native. Non-native problems can only be transformed into MinCostSat by adding slack variables. These problems include the Max-Sat, partial Max-Sat, and group-partial Max-Sat problems which have applications ranging from course assignment to FPGA detailed routing. Native problems are various sub-cases of MinCostSat. We further divide these into two

The Boolean vector function synthesis problem can be stated as follows: Given a truth table with n input variables and m output variables, synthesize a Boolean vector function that describes the table. In this paper we describe a new formulation of the Boolean vector function synthesis problem as a particular type of Satisfiability Problem. The Satisfiability Problem is translated into an integer programming feasibility problem, that is solved with an interior point algorithm for integer programming. Preliminary computational results are presented. Introduction The Boolean Vector Function Synthesis Problem has applications in logic, artificial intelligence, machine learning, and digital integrated circuit design. In this paper, we describe a Satisfiability Problem formulation of the Boolean Vector Function Synthesis Problem. This formulation can be approached with a wide range of algorithms. In this paper, preliminary computational results are presented using an interior point algorit...

The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region problem by allowing a general quadratic constraint. The main results are a characterization of the global minimizer of the generalized trust region problem, and the development of an algorithm that finds an approximate global minimizer in a finite number of iterations.

We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to `real-life' problems show that the algorithm is competitive with the fastest algorithms known so far. 1 Introduction We denote by G = (V; E) an undirected graph. V is the set of nodes and E the set of edges. A stable set is a subset of V such that no two nodes of the subset are pairwise adjacent. The cardinality of a maximum stable set of G will be denoted by ff(G). A clique is a subset of V with the property that all the nodes are pairwise adjacent. A clique covering is a set of disjoint cliques whose union is equal to V ; the cardinality of a minimum clique covering is denoted by `(G), and since at most one nod...

The frequency assignment problem is the problem of assigning frequencies to transmission links such that either no interference occurs and the number of used frequencies is minimized, or the amount of interference is minimized. We present an algorithm for this problem that is based on Karmarkar's interior point potential reduction approach to combinatorial optimization problems. We develop a new quadratic formulation of the problem that reduces the problem size significantly. Several preprocessing techniques are discussed. We report on computational experience with real-life instances, applying Karmarkar's algorithm to the quadratic model. Acknowledgement This report completes my study of Technical Mathematics at the Delft University of Technology. During the period April 1994 to March 1995 I have been working on the development of an interior point potential reduction algorithm for the radio link frequency assignment problem. During these months a number of people were of great help....