pseudo-Boolean Functions Research Articles

Short Paper - The Binary Linearization Complexity of Pseudo-Boolean Functions

Short Paper - The Binary Linearization Complexity of Pseudo-Boolean Functions

Characterization of rankings generated by pseudo-Boolean functions

In this paper we pursue the study of pseudo-Boolean functions as ranking generators. The objective of the work is to find new insights between the relation of the degree m of a pseudo-Boolean function and the rankings that can be generated by these insights. Based on a characterization theorem for pseudo-Boolean functions of degree m, several observations are made. First, we verify that pseudo-Boolean functions of degree m<n, where n is the search space dimension, cannot generate all the possible rankings of the solutions. Secondly, the sufficient condition for a ranking to be generated by a pseudo-Boolean function of dimension (n−1) is presented, and also the necessary condition is conjectured. Finally, we observe that the same argument is not sufficient to prove which ranking can be generated by pseudo-Boolean functions of degree m<n−1.

Discovering Non-Linear Boolean Functions by Evolving Walsh Transforms with Genetic Programming

Stream ciphers usually rely on highly secure Boolean functions to ensure safe communication within unsafe channels. However, discovering secure Boolean functions is a non-trivial optimization problem that has been addressed by many optimization techniques: in particular by evolutionary algorithms. We investigate in this article the employment of Genetic Programming (GP) for evolving Boolean functions with large non-linearity by examining the search space consisting of Walsh transforms. Especially, we build generic Walsh spectra starting from the evolution of Walsh transform coefficients. Then, by leveraging spectral inversion, we build pseudo-Boolean functions from which we are able to determine the corresponding nearest Boolean functions, whose computation involves filling via a random criterion a certain amount of “uncertain” positions in the final truth table. We show that by using a balancedness-preserving strategy, it is possible to exploit those positions to obtain a function that is as balanced as possible. We perform experiments by comparing different types of symbolic representations for the Walsh transform, and we analyze the percentage of uncertain positions. We systematically review the outcomes of these comparisons to highlight the best type of setting for this problem. We evolve Boolean functions from 6 to 16 bits and compare the GP-based evolution with random search to show that evolving Walsh transforms leads to highly non-linear functions that are balanced as well.

Feature Selection via Nonlinear Set Covering in Logical Analysis of Data

In logical analysis of data, the feature selection step only considers selecting minimal number of features after binarization. This paper develops a nonlinear set covering model that explores the interaction between the number of selected original attributes and binarized features. Utilizing the partial derivative of pseudo-Boolean functions, we give a greedy heuristic for the nonlinear set covering problem. The efficacy of the algorithm is demonstrated through experiments on 10 public machine learning datasets from UCI repository.

Total-Activity Conservation Analysis and Design of Boolean Networks.

The law of conservation of mass, represented in Boolean networks (BNs) as total-activity conservation, is one of the typical properties of biological networks. This article analyzes the total-activity conservation of BNs based on the algebraic state-space representation (ASSR) approach. First, the total-activity-conservative matrix is defined and a matrix-based criterion is proposed to verify the total-activity conservation of BNs. Meanwhile, when function perturbation is considered, robust total-activity conservation is investigated. Second, by means of the pseudo-Boolean function generated by total-activity conservation, a constructive design procedure of the Boolean dynamics is given to achieve the total-activity conservation. Third, the total-activity conservation of switched Boolean networks (SBNs) under arbitrary switching signal is studied, which together with network aggregation achieves the total-activity conservation of large-scale BNs.

Symmetry Breaking for Voting Mechanisms.

Recently, Rowe and Aishwaryaprajna (2019) introduced a simple majority vote technique that efficiently solves Jump with large gaps, OneMax with large noise, and any monotone function with a polynomial-size image. In this paper, we identify a pathological condition for this algorithm: the presence of spin-flip symmetry in the problem instance. Spin-flip symmetry is the invariance of a pseudo-Boolean function to complementation. Many important combinatorial optimization problems admit objective functions that exhibit this pathology, such as graph problems, Ising models, and variants of propositional satisfiability. We prove that no population size exists that allows the majority vote technique to solve spin-flip symmetric functions of unitation with reasonable probability. To remedy this, we introduce a symmetry-breaking technique that allows the majority vote algorithm to overcome this issue for many landscapes. This technique requires only a minor modification to the original majority vote algorithm to force it to sample strings in {0,1}n from a dimension n-1 hyperplane. We prove a sufficient condition for a spin-flip symmetric function to possess in order for the symmetry-breaking voting algorithm to succeed, and prove its efficiency on generalized TwoMax, a spin-flip symmetric variant of Jump, and families of constructed 3-NAE-SAT and 2-XOR-SAT formulas. We also prove that the algorithm fails on the one-dimensional Ising model, and suggest different techniques for overcoming this. Finally, we present empirical results that explore the tightness of the runtime bounds and the performance of the technique on randomized satisfiability variants.

Crossover for Cardinality Constrained Optimization

To understand better how and why crossover can benefit constrained optimization, we consider pseudo-Boolean functions with an upper bound B on the number of 1-bits allowed in the length- n bit string (i.e., a cardinality constraint). We investigate the natural translation of the OneMax test function to this setting, a linear function where B bits have a weight of 1+ 1/ n and the remaining bits have a weight of 1. Friedrich et al. [TCS 2020] gave a bound of Θ ( n 2 ) for the expected running time of the (1+1) EA on this function. Part of the difficulty when optimizing this problem lies in having to improve individuals meeting the cardinality constraint by flipping a 1 and a 0 simultaneously. The experimental literature proposes balanced operators, preserving the number of 1-bits, as a remedy. We show that a balanced mutation operator optimizes the problem in O(n log n ) if n-B = O (1). However, if n-B = Θ ( n ), we show a bound of Ω ( n 2 ), just as for classic bit mutation. Crossover together with a simple island model gives running times of O ( n 2 / log n ) (uniform crossover) and \(O(n\sqrt {n})\) (3-ary majority vote crossover). For balanced uniform crossover with Hamming-distance maximization for diversity, we show a bound of O ( n log n ). As an additional contribution, we present an extensive analysis of different balanced crossover operators from the literature.

Walsh-based surrogate-assisted multi-objective combinatorial optimization: A fine-grained analysis for pseudo-boolean functions

The aim of this paper is to study surrogate-assisted algorithms for expensive multiobjective combinatorial optimization problems. Targeting pseudo-boolean domains, we provide a fine-grained analysis of an optimization framework using the Walsh basis as a core surrogate model. The considered framework uses decomposition in the objective space, and integrates three different components, namely, (i) an inner optimizer for searching promising solutions with respect to the so-constructed surrogate, (ii) a selection strategy to decide which solution is to be evaluated by the expensive objectives, and (iii) the strategy used to setup the Walsh order hyper-parameter. Based on extensive experiments using two benchmark problems, namely bi-objective NK-landscapes and unconstrained binary quadratic programming problems (UBQP), we conduct a comprehensive in-depth analysis of the combined effects of the considered components on search performance, and provide evidence on the effectiveness of the proposed search strategies. As a by-product, our work shed more light on the key challenges for designing a successful surrogate-assisted multi-objective combinatorial search process.

On the Use of Gray Codes for Solving 0-1 Combinatorial Problems of Optimization in Environmental and Economic Systems

Introduction. The application of binary-reflected (mirror, reflexive) Gray codes for solving combinatorial problems with pseudo-Boolean functions (polynomials from Boolean variables) is considered. A recursive Ehrlich algorithm is given for generating a sequence of lines n-bit Gray codes, in which each subsequent line differs from the previous one by only one digit (bit). As an example of the effectiveness of the use of these codes, the solution of two combinatorial problems with Boolean variables with a complete enumeration of solutions is considered, and it is shown how these codes can be used to efficiently calculate the values of the objective function and constraints. The results of an experimental study are presented, which show that Gray codes can be practically applied in branching schemes, for example, in the branch and bound method, when the number of variables in the branching nodes of the decision algorithm does not exceed 35. Purpose. The purpose of the article is to show the developers of algorithms and programs how to apply Gray codes in various branching schemes of the decision algorithm, for example, in the branch and bound method, when the number of binary (Boolean) variables at the nodes of the tree is small (less than 35). The technique. The research methodology is based on a computational experiment for solving the 0-1 knapsack problem with the proposed algorithm of exhaustive search the solution with partial and full recalculation of the values of objective function and constraint of the problem. During the experiment, the accuracy of solving the problem by a “greedy” heuristic algorithm with time complexity O(n2) was also checked. Results. As a result of the experiment, it was found that the algorithm with a partial recalculation of the objective function and restrictions can be used for practical calculations in branching schemes, when the number of variables in the nodes of the branching tree does not exceed 35. The algorithm with partial recalculation is faster than the algorithm with full recalculation on average by 7 times. The heuristic "greedy" algorithm can be applied in practice to solve the 0-1 problem of a knapsack of large dimension (more than 10,000 items), when need to obtain an approximate value of the objective function at the limited computing resources. Scientific novelty and practical significance. The novelty of the work lies in the proposed approach to solving combinatorial optimization problems with pseudo-Boolean functions using Gray codes. The efficiency of the proposed algorithm with a partial recalculation of the values of the objective function and constraints is shown, and its can be applied in practice in various branching schemes of the decision algorithm. Keywords: Gray codes, combinatorial optimization problems, problem solving time.

On the compounding of higher order monotonic pseudo-Boolean functions

Compounding submodular monotone (i.e. 2-alternating) set functions on a finite set preserves this property, as shown in 2010. A natural generalization to k-alternating functions was presented in 2018, however hardly readable because of page long formulas. We give an easier proof of a more general result, exploiting known properties of higher order monotonic functions.

Direct Construction of Optimal Z-Complementary Code Sets With Even Lengths by Using Generalized Boolean Functions

Z-complementary code set (ZCCS) are well known to be used in multicarrier code-division multiple access (MCCDMA) system to provide a interference free environment. Based on the existing literature, the direct construction of optimal ZCCSs are limited to its length. In this paper, we are interested in constructing optimal ZCCSs of all possible even lengths using Pseudo-Boolean functions. The maximum column sequence peakto-man envelop power ratio (PMEPR) of the proposed ZCCSs is upper-bounded by two, which may give an extra benefit in managing PMEPR in an ZCCS based MC-CDMA system, as well as the ability to handle a large number of users.

Modeling the dynamics of a changing range genetic algorithm in noisy environments

The paper discussed the dynamic model of a genetic algorithm (GA) with an adjustment of a search space size in a noisy environment. A linear pseudo-Boolean function with positive coefficients was used for modeling the GA with a search space size adjustment. The impact of an additive noise on the fitness value of the GA is assessed, and analytical expressions are derived that described how GA’s selection and size adjustment operators alter the macroscopic statistical properties of population. Calculations based on the theory are compared with experiments demonstrating the accurate predictions for the macroscopic statistical properties of population.

A Novel Approach to Designing Surrogate-assisted Genetic Algorithms by Combining Efficient Learning of Walsh Coefficients and Dependencies

Surrogate-assisted evolutionary algorithms have the potential to be of high value for real-world optimization problems when fitness evaluations are expensive, limiting the number of evaluations that can be performed. In this article, we consider the domain of pseudo-Boolean functions in a black-box setting. Moreover, instead of using a surrogate model as an approximation of a fitness function, we propose to precisely learn the coefficients of the Walsh decomposition of a fitness function and use the Walsh decomposition as a surrogate. If the coefficients are learned correctly, then the Walsh decomposition values perfectly match with the fitness function, and, thus, the optimal solution to the problem can be found by optimizing the surrogate without any additional evaluations of the original fitness function. It is known that the Walsh coefficients can be efficiently learned for pseudo-Boolean functions with k -bounded epistasis and known problem structure. We propose to learn dependencies between variables first and, therefore, substantially reduce the number of Walsh coefficients to be calculated. After the accurate Walsh decomposition is obtained, the surrogate model is optimized using GOMEA, which is considered to be a state-of-the-art binary optimization algorithm. We compare the proposed approach with standard GOMEA and two other Walsh decomposition-based algorithms. The benchmark functions in the experiments are well-known trap functions, NK-landscapes, MaxCut, and MAX3SAT problems. The experimental results demonstrate that the proposed approach is scalable at the supposed complexity of O (ℓ log ℓ) function evaluations when the number of subfunctions is O (ℓ) and all subfunctions are k -bounded, outperforming all considered algorithms.

Pseudo-Boolean Functions for Optimal Z-Complementary Code Sets With Flexible Lengths

This paper aims to construct optimal Z-complementary code set (ZCCS) with non-power-of-two (NPT) lengths to enable interference-free multicarrier code-division multiple access (MC-CDMA) systems. The existing ZCCSs with NPT lengths, which are constructed from generalized Boolean functions (GBFs), are sub-optimal only with respect to the set size upper bound. For the first time in the literature, we advocate the use of pseudo-Boolean functions (PBFs) (each of which transforms a number of binary variables to a real number as a natural generalization of GBF) for direct constructions of optimal ZCCSs with NPT lengths.

Frequency‐constrained unit‐commitment using analytical solutions for system frequency responses considering generator contingencies

The frequency responses of a power system following a generator outage are closely related to the commitment states of generators, which describe the droop and inertial characteristics and system pre-contingency power dispatch. This study presents a new model to estimate the transient and steady state frequency responses of two-area power systems at any moment following the loss of an online generator. In the proposed model, the deviations of tie-flow, area frequencies and rate of change of area frequencies as well as the extreme values of these functions and their associated time of occurrence following the outage of a generation unit are derived as modified analytical functions. A new approach is also proposed to linearise the modified extreme deviations based on pseudo-Boolean function theorem. In addition, a frequency-constrained unit commitment model with modified frequency-dependent constraints incorporating primary frequency regulation reserve is presented as a mixed-integer linear programming problem. The results indicate that the proposed method gives acceptable approximations for post-contingency responses and can ensure power system security from frequency viewpoint.

Matrix Expression of Shapley Value in Graphical Cooperative Games

This paper studies a class of cooperative games, called graphical cooperative games, where the internal topology of the coalition depends on a prescribed communication graph among players. First, using the semitensor product of matrices, the value function of graphical cooperative games can be expressed as a pseudo-Boolean function. Then, a simple matrix formula is provided to calculate the Shapley value of graphical cooperative games. Finally, some practical examples are presented to illustrate the application of graphical cooperative games in communication-based coalitions and establish the significance of the Shapley value in different communication networks.

Compact quadratizations for pseudo-Boolean functions

The problem of minimizing a pseudo-Boolean function, that is, a real-valued function of 0–1 variables, arises in many applications. A quadratization is a reformulation of this nonlinear problem into a quadratic one, obtained by introducing a set of auxiliary binary variables. A desirable property for a quadratization is to introduce a small number of auxiliary variables. We present upper and lower bounds on the number of auxiliary variables required to define a quadratization for several classes of specially structured functions, such as functions with many zeros, symmetric, exact k-out-of-n, at least k-out-of-n and parity functions, and monomials with a positive coefficient, also called positive monomials. Most of these bounds are logarithmic in the number of original variables, and we prove that they are best possible for several of the classes under consideration. For positive monomials and for some other symmetric functions, a logarithmic bound represents a significant improvement with respect to the best bounds previously published, which are linear in the number of original variables. Moreover, the case of positive monomials is particularly interesting: indeed, when a pseudo-Boolean function is represented by its unique multilinear polynomial expression, a quadratization can be obtained by separately quadratizing its monomials.

Running Time Analysis of MOEA/D on Pseudo-Boolean Functions.

Decomposition-based multiobjective evolutionary algorithms (MOEAs) are a class of popular methods for solving the multiobjective optimization problems (MOPs), and have been widely studied in numerical experiments and successfully applied in practice. However, we know little about these algorithms from the theoretical aspect. In this paper, we present running time analysis of a simple MOEA with mutation and crossover based on the MOEA/D framework (MOEA/D-C) on five pseudo-Boolean functions. Our rigorous theoretical analysis shows that by properly setting the number of subproblems, the upper bounds of expected running time of MOEA/D-C obtaining a set of solutions to cover the Pareto fronts (PFs) of these problems are apparently lower than those of the one with mutation-only (MOEA/D-M). Moreover, to effectively obtain a set of solutions to cover the PFs of these problem, MOEA/D-C only needs to decompose these MOPs into several subproblems with a set of simple weight vectors while MOEA/D-M needs to find Ω(n) optimally decomposed weight vectors. This result suggests that the use of crossover in decomposition-based MOEA can simplify the setting of weight vectors for different problems and make the algorithm more efficient. This paper provides some insights into the working principles of MOEA/D and explains why some existing decomposition-based MOEAs work well in computational experiments.

Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions

Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.

Matrix expression of Shapley values and its application to distributed resource allocation

The symmetric and weighted Shapley values for cooperative n-person games are studied. Using the semi-tensor product of matrices, it is first shown that a characteristic function can be expressed as a pseudo-Boolean function. Then, two simple matrix formulas are obtained for calculating the symmetric and weighted Shapley values. Finally, using these new formulas, a design technique for the agents’ payoff functions in distributed resource allocation problems is proposed. It is possible to design payoff functions with the weighted Shapley value by the nonsymmetric weights defined on the players, thus ensuring that the optimal allocation is a pure Nash equilibrium. Practical examples are presented to illustrate the theoretical results.