Boolean Functions: Theory, Algorithms, and Applications

This monograph provides the first comprehensive presentation of the theoretical, algorithmic and applied aspects of Boolean functions, i.e., {0,1}-valued functions of a finite number of {0,1}-valued variables. The book focuses on algebraic representations of Boolean functions, especially normal form representations. It presents the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated representations, dualization, etc.), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once, etc.), and two fruitful generalizations of the concept of Boolean functions (partially defined and pseudo-Boolean functions). It features a rich bibliography of about one thousand items. Prominent among the disciplines in which Boolean methods play a significant role are propositional logic, combinatorics, graph and hypergraph theory, complexity theory, integer programming, combinatorial optimization, game theory, reliability theory, electrical and computer engineering, artificial intelligence, etc. The book contains applications of Boolean functions in all these areas

Matroids arisen from matrogenic graphs

Let G be a finite simple graph and let ℐ(G) be the set of subsets X of V(G) such that the subgraph of G induced by X is threshold. If ℐ(G) is the independence system of a matroid, then G is called matrogenic [3]. In this paper, we characterize matroids arising from matrogenic graphs

Spanned patterns for the logical analysis of data

AbstractIn a finite dataset consisting of positive and negative observations represented as real valued n-vectors, a positive (negative) pattern is an interval in Rn with the property that it contains sufficiently many positive (negative) observations, and sufficiently few negative (positive) ones. A pattern is spanned if it does not include properly any other interval containing the same set of observations. Although large collections of spanned patterns can provide highly accurate classification models within the framework of the Logical Analysis of Data, no efficient method for their generation is currently known. We propose in this paper, an incrementally polynomial time algorithm for the generation of all spanned patterns in a dataset, which runs in linear time in the output; the algorithm resembles closely the Blake and Quine consensus method for finding the prime implicants of Boolean functions. The efficiency of the proposed algorithm is tested on various publicly available datasets. In the last part of the paper, we present the results of a series of computational experiments which show the high degree of robustness of spanned patterns

Maximally Disjoint Solutions of the Set Covering Problem

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP­ complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, extensive computational tests are given demonstrating the nature of these algorithms. These tests are carried out both on randomly generated problems and on problems found in the literature

Identification And Management Of Pregnancy‐Related Low Back Pain

Back pain is a common complaint of women during pregnancy. It is frequently dismissed as trivial and inevitable. This article gives an overview of recent research on pregnancy‐related back pain that documents the impact of this pain on women's lives, during and beyond the childbearing year. It argues for a more active approach to the prevention and management of back pain during pregnancy. Two common back pain types, lumbar pain and posterior pelvic pain, are described and basic management techniques for the woman and her primary caregiver are suggested. Red flag findings that necessitate specialist referral are also highlighted, as are suggestions for further research.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90043/1/j.1542-2011.1998.tb03313.x.pd

Product form parametric representation of the solutions to a quadratic boolean equation

A parametric représentation of the solutions to a consistent quadratic boolean equation in n variables is obtained. Each variable (or its complement) is expressed as a product of free boolean parameters or their complements. These expressions provide a complete description of the solution set of the equation. An O (n^3) algorithm is proposed to produce such a representation. An application to the maximization of some classes of pseudoboolean functions is discussed

A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO)

AbstractThe “roof dual” of a QUBO (Quadratic Unconstrained Binary Optimization) problem has been introduced in [P.L. Hammer, P. Hansen, B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Mathematical Programming 28 (1984) 121–155]; it provides a bound to the optimum value, along with a polynomial test of the sharpness of this bound, and (due to a “persistency” result) it also determines the values of some of the variables at the optimum. In this paper we provide a graph-theoretic approach to provide bounds, which includes as a special case the roof dual bound, and show that these bounds can be computed in O(n3) time by using network flow techniques. We also obtain a decomposition theorem for quadratic pseudo-Boolean functions, improving the persistency result of [P.L. Hammer, P. Hansen, B. Simeone, Roof duality, complementation and persistency in quadratic 0–1 optimization, Mathematical Programming 28 (1984) 121–155]. Finally, we show that the proposed bounds (including roof duality) can be applied in an iterated way to obtain significantly better bounds. Computational experiments on problems up to thousands of variables are presented

ATHENA: A Phase 3, Open-Label Study Of The Safety And Effectiveness Of Oliceridine (TRV130), A G-Protein Selective Agonist At The µ-Opioid Receptor, In Patients With Moderate To Severe Acute Pain Requiring Parenteral Opioid Therapy.

Background:Pain management with conventional opioids can be challenging due to dose-limiting adverse events (AEs), some of which may be related to the simultaneous activation of β-arrestin (a signaling pathway associated with opioid-related AEs) and G-protein pathways. The investigational analgesic oliceridine is a G-protein-selective agonist at the µ-opioid receptor with less recruitment of β-arrestin. The objective of this phase 3, open-label, multi-center study was to evaluate the safety and tolerability, of IV oliceridine for moderate to severe acute pain in a broad, real-world patient population, including postoperative surgical patients and non-surgical patients with painful medical conditions. Methods:Adult patients with a score ≥4 on 11-point NRS for pain intensity received IV oliceridine either by bolus or PCA; multimodal analgesia was permitted. Safety was assessed using AE reports, study discontinuations, clinical laboratory and vital sign measures. Results:A total of 768 patients received oliceridine. The mean age (SD) was 54.1 (16.1) years, with 32% ≥65 years of age. Most patients were female (65%) and Caucasian (78%). Surgical patients comprised the majority of the study population (94%), most common being orthopedic (30%), colorectal (15%) or gynecologic (15%) procedures. Multimodal analgesia was administered to 84% of patients. Oliceridine provided a rapid reduction in NRS pain score by 2.2 ± 2.3 at 30 mins from a score of 6.3 ± 2.1 (at baseline) which was maintained to the end of treatment. No deaths or significant cardiorespiratory events were reported. The incidence of AEs leading to early discontinuation and serious AEs were 2% and 3%, respectively. Nausea (31%), constipation (11%), and vomiting (10%) were the most common AEs. AEs were mostly of mild (37%) or moderate (25%) severity and considered possibly or probably related to oliceridine in 33% of patients. Conclusion:Oliceridine IV for the management of moderate to severe acute pain was generally safe and well tolerated in the patients studied. ClinicalTrialsgov identifier:NCT02656875

Breast cancer prognosis by combinatorial analysis of gene expression data

INTRODUCTION: The potential of applying data analysis tools to microarray data for diagnosis and prognosis is illustrated on the recent breast cancer dataset of van 't Veer and coworkers. We re-examine that dataset using the novel technique of logical analysis of data (LAD), with the double objective of discovering patterns characteristic for cases with good or poor outcome, using them for accurate and justifiable predictions; and deriving novel information about the role of genes, the existence of special classes of cases, and other factors. METHOD: Data were analyzed using the combinatorics and optimization-based method of LAD, recently shown to provide highly accurate diagnostic and prognostic systems in cardiology, cancer proteomics, hematology, pulmonology, and other disciplines. RESULTS: LAD identified a subset of 17 of the 25,000 genes, capable of fully distinguishing between patients with poor, respectively good prognoses. An extensive list of 'patterns' or 'combinatorial biomarkers' (that is, combinations of genes and limitations on their expression levels) was generated, and 40 patterns were used to create a prognostic system, shown to have 100% and 92.9% weighted accuracy on the training and test sets, respectively. The prognostic system uses fewer genes than other methods, and has similar or better accuracy than those reported in other studies. Out of the 17 genes identified by LAD, three (respectively, five) were shown to play a significant role in determining poor (respectively, good) prognosis. Two new classes of patients (described by similar sets of covering patterns, gene expression ranges, and clinical features) were discovered. As a by-product of the study, it is shown that the training and the test sets of van 't Veer have differing characteristics. CONCLUSION: The study shows that LAD provides an accurate and fully explanatory prognostic system for breast cancer using genomic data (that is, a system that, in addition to predicting good or poor prognosis, provides an individualized explanation of the reasons for that prognosis for each patient). Moreover, the LAD model provides valuable insights into the roles of individual and combinatorial biomarkers, allows the discovery of new classes of patients, and generates a vast library of biomedical research hypotheses