Endre Boros

Pseudo-boolean optimization

This survey examines the state of the art of a variety of problems related to pseudo-Boolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudo-Boolean optimization, the specially important case of quadratic pseudo-Boolean optimization (to which every pseudo-Boolean optimization can be reduced), several other important special classes, and approximation algorithms.

Logical analysis of numerical data

Abstract“Logical analysis of data” (LAD) is a methodology developed since the late eighties, aimed at discovering hidden structural information in data sets. LAD was originally developed for analyzing binary data by using the theory of partially defined Boolean functions. An extension of LAD for the analysis of numerical data sets is achieved through the process of “binarization” consisting in the replacement of each numerical variable by binary “indicator” variables, each showing whether the value of the original variable is above or below a certain level. Binarization was successfully applied to the analysis of a variety of real life data sets. This paper develops the theoretical foundations of the binarization process studying the combinatorial optimization problems related to the minimization of the number of binary variables. To provide an algorithmic framework for the practical solution of such problems, we construct compact linear integer programming formulations of them. We develop polynomial time algorithms for some of these minimization problems, and prove NP-hardness of others.

Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur

In two previous papers Prekopa (Prekopa, A. 1986a. Boole-Bonferroni inequalities and linear programming. Oper. Res. 36 145–162; Prekopa, A. 1986b. Sharp bounds on probabilities using linear programming. To appear in Oper. Res.) gave algorithms to approximate probabilities that at least r and exactly r out of n events occur (1 ≤ r ≤ n). Primal and dual linear programming problems were formulated and solved by dual type algorithms. The purpose of the present paper is to give closed forms for the basis inverse and the corresponding dual vector in case of an arbitrary basis, furthermore to give closed forms for the lower and upper bounds, approximating the probability in question, in case of a dual feasible basis. In the case when the probability that at least one out of n events occurs is approximated, it is shown that the absolute values of the components of any dual vector form a monotonically decreasing sequence. The paper improves the method of inclusion-exclusion, proves new probability inequalities and...

On the Complexity of Generating Maximal Frequent and Minimal Infrequent Sets

Let A be an m × n binary matrix, t ? {1, ..., m} be a threshold, and ? > 0 be a positive parameter. We show that given a family of O(n?) maximal t-frequent column sets for A, it is NP-complete to decide whether A has any further maximal t-frequent sets, or not, even when the number of such additional maximal t-frequent column sets may be exponentially large. In contrast, all minimal t-infrequent sets of columns of A can be enumerated in incremental quasi-polynomial time. The proof of the latter result follows from the inequality ? ? (m-t+1)s, where ? and s are respectively the numbers of all maximal t-frequent and all minimal t-infrequent sets of columns of the matrix A. We also discuss the complexity of generating all closed t-frequent column sets for a given binary matrix.

Polynomial-time inference of all valid implications for Horn and related formulae

This paper investigates the complexity of a general inference problem: given a propositional formula in conjunctive normal form, find all prime implications of the formula. Here, a prime implication means a minimal clause whose validity is implied by the validity of the formula. We show that, under some reasonable assumptions, this problem can be solved in time polynomially bounded in the size of the input and in the number of prime implications. In the case of Horn formulae, the result specializes to yield an algorithm whose complexity grows only linearly with the number of prime implications. The result also applies to a class of formulae generalizing both Horn and quadratic formulae.

The number of triangles covering the center of an n-set

Let the points P1, P2, ..., Pnbe given in the plane such that there are no three on a line. Then there exists a point of the plane which is contained in at least n3/27 (open) PiPjPktriangles. This bound is the best possible.

Error-free and best-fit extensions of partially defined Boolean functions

Abstract In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.

Recognition of q -Horn formulae in linear time

Abstract The class of q-Horn Boolean expressions, generalizing the important classes of quadratic, Horn, and disguised Horn formulae, has been introduced in Boros et al.(1990). It has been shown there that the satisfiability problem corresponding to a disjunctive normal form φ is solvable in time, linear in the size of φ, if φ is known to be q-Horn. However, the recognition of such formulae was based on the solution of a linear programming problem, and had therefore a much higher (although still polynomial) complexity. In this paper a linear-time combinatorial algorithm is presented for recognizing q-Horn formulae, and reducing in this way the overall complexity of the corresponding satisfiability problem to a linear one.

The max-cut problem and quadratic 0–1 optimization; polyhedral aspects, relaxations and bounds

Given a graphG, themaximum cut problem consists of finding the subsetS of vertices such that the number of edges having exactly one endpoint inS is as large as possible. In the weighted version of this problem there are given real weights on the edges ofG, and the objective is to maximize the sum of the weights of the edges having exactly one endpoint in the subsetS. In this paper, we consider the maximum cut problem and some related problems, likemaximum-2-satisfiability, weighted signed graph balancing. We describe the relation of these problems to the unconstrained quadratic 0–1 programming problem, and we survey the known methods for lower and upper bounds to this optimization problem. We also give the relation between the related polyhedra, and we describe some of the known and some new classes of facets for them.

A graph cut algorithm for higher-order Markov Random Fields

Higher-order Markov Random Fields, which can capture important properties of natural images, have become increasingly important in computer vision. While graph cuts work well for first-order MRFs, until recently they have rarely been effective for higher-order MRFs. Ishikawas graph cut technique [8, 9] shows great promise for many higher-order MRFs. His method transforms an arbitrary higher-order MRF with binary labels into a first-order one with the same minima. If all the terms are submodular the exact solution can be easily found; otherwise, pseudo-boolean optimization techniques can produce an optimal labeling for a subset of the variables. We present a new transformation with better performance than [8, 9], both theoretically and experimentally. While [8, 9] transforms each higher-order term independently, we transform a group of terms at once. For n binary variables, each of which appears in terms with k other variables, at worst we produce n non-submodular terms, while [8, 9] produces O(nk). We identify a local completeness property that makes our method perform even better, and show that under certain assumptions several important vision problems (including common variants of fusion moves) have this property. Running on the same field of experts dataset used in [8, 9] we optimally label significantly more variables (96% versus 80%) and converge more rapidly to a lower energy. Preliminary experiments suggest that some other higher-order MRFs used in stereo [20] and segmentation [1] are also locally complete and would thus benefit from our work.