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Characterization of rankings generated by pseudo-boolean functions

SWARM AND EVOLUTIONARY COMPUTATION 2024年 86卷

作者： Unanue, Imanol Merino, Maria Lozano, Jose A. Univ Basque Country UPV EHU Dept Comp Sci & Artificial Intelligence Manuel Lardizabal Pasealekua 1 Donostia San Sebastian 20018 Spain Univ Basque Country UPV EHU Dept Math Leioa 48940 Spain BCAM Basque Ctr Appl Math Mazarredo Zumarkalea 14 Bilbao 48009 Spain

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 .

关键词： Binary-based combinatorial optimization problems pseudo-boolean functions Rankings Dyck Theoretical analysis

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Compact quadratizations for pseudo-boolean functions

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JOURNAL OF COMBINATORIAL OPTIMIZATION 2020年第3期39卷 687-707页

作者： Boros, Endre Crama, Yves Rodriguez-Heck, Elisabeth Rutgers State Univ MSIS Dept Piscataway NJ USA Rutgers State Univ RUTCOR Piscataway NJ USA Univ Liege HEC Liege QuantOM Liege Belgium Rhein Westfal TH Aachen Aachen Germany

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.

关键词： Nonlinear binary optimization Quadratic binary optimization pseudo-boolean functions Reformulation methods

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Understanding Transforms of pseudo-boolean functions

Understanding Transforms of Pseudo-Boolean Functions

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Whitley, Darrell Aguirre, Hernan Sutton, Andrew Colorado State Univ Ft Collins CO 80523 USA Shinshu Univ Nagano Japan Univ Minnesota Duluth MN 55812 USA

ISBN: (纸本)9781450371285

There exist general transforms that convert pseudo-boolean functions into k-bounded pseudo-boolean functions, for all k >= 2. In addition to these general transforms, there can also exist specialized transforms that can be applied in special cases. New results are presented examining what happens to the "bit flip" neighborhood when transforms are applied. Transforms condense variables in a particular order. We show that different variable orderings produce different results in terms of problem difficulty. We also prove new results about the embedding of the original function in the new k-bounded function. Finally, this paper also looks at how parameter optimization problems can be expressed as high precision k-bounded pseudo-boolean functions. This paper lays a foundation for the wider application of evolutionary algorithms to k-bounded pseudo-boolean functions.

关键词： combinatorial optimization pseudo-boolean functions bit representations epistatsis

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Learning and Searching pseudo-boolean Surrogate functions from Small Samples

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EVOLUTIONARY COMPUTATION 2020年第2期28卷 317-338页

作者： Swingler, Kevin Univ Stirling Comp Sci & Math Stirling FK9 4LA Scotland

When searching for input configurations that optimise the output of a system, it can be useful to build a statistical model of the system being optimised. This is done in approaches such as surrogate model-based optimisation, estimation of distribution algorithms, and linkage learning algorithms. This article presents a method for modelling pseudo-boolean fitness functions using Walsh bases and an algorithm designed to discover the non-zero coefficients while attempting to minimise the number of fitness function evaluations required. The resulting models reveal linkage structure that can be used to guide a search of the model efficiently. It presents experimental results solving benchmark problems in fewer fitness function evaluations than those reported in the literature for other search methods such as EDAs and linkage learners.

关键词： Fitness function modelling estimation of distribution algorithms pseudo-boolean functions linkage learning Walsh decomposition mixed order hyper networks statistical machine learning

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Quadratization of Gray Coded Representations, Long Path Problems and Needle functions

Quadratization of Gray Coded Representations, Long Path Prob...

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2nd Genetic and Evolutionary Computation Conference (GECCO)

作者： Whitley, Darrell Chicano, Francisco Aguirre, Hernan Colorado State Univ Ft Collins CO 80523 USA Univ Malaga Malaga Spain Shinshu Univ Nagano Japan

ISBN: (纸本)9781450383509

In Evolutionary Computation, it is informative to ask what happens when well known benchmarks and bit representations are transformed into quadratic pseudo-boolean optimization problems. Such transforms are commonly used in quantum computing in order to reduce nonlinearity to k-bounded interactions. First we show that Gray code representations are transformed into linear encodings with quadratic constraints. Next we look at Long Path problems which are constructed so that bit flip local search requires exponential time to converge to a global or local optimum. We show that Long Path problems are similar to reflected Gray codes in both construction and complexity. Finally, a basic form of the "Needle in a haystack" problem is transformed into a problem that can be optimally solved in linear time.

关键词： Quadratization pseudo-boolean functions Representations Gray Codes

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Understanding the Cost of Fitness Evaluation for Subset Selection: Markov Chain Analysis of Stochastic Local Search

Understanding the Cost of Fitness Evaluation for Subset Sele...

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Genetic and Evolutionary Computation Conference (GECCO)

作者： Mengshoel, Ole Jakob Flogard, Eirik Lund Yu, Tong Riege, Jon Norwegian Univ Sci & Technol Trondheim Norway Norwegian Labour Inspect Author Trondheim Norway Carnegie Mellon Univ Pittsburgh PA 15213 USA Boston Consulting Grp Inc Oslo Norway

ISBN: (纸本)9781450392372

With a focus on both the fitness and cost of subset selection, we study stochastic local search (SLS) heuristics in this paper. In particular, we consider subset selection problems where the cost of fitness function evaluation needs to be accounted for. Here, cost can be fitness evaluation's computation time or energy cost. We propose and study an SLS method, SLS4CFF, tailored to such problems. SLS4CFF ("SLS for costly fitness functions") is an amalgamation of several existing heuristics. We develop a homogeneous Markov chain model that explicitly represents both fitness and cost of subset selection with SLS4CFF. This Markov chain, which can be lumped or compressed for certain fitness and cost functions, enables us to better understand and analyze hyperparameter optimization in a principled manner, via expected hitting times. Studies with synthetic and real-world problems improve the understanding of SLS and demonstrate the importance of cost-awareness.

关键词： pseudo-boolean functions costly fitness functions synthetic problems optimization stochastic local search Markov chain analysis expected hitting time machine learning classification

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On anonymous and weighted voting systems

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THEORY AND DECISION 2021年第4期91卷 477-491页

作者： Freixas, Josep Pons, Montserrat Univ Politecn Cataluna Dept Math EPSEM Campus ManresaAv Bases Manresa 61-73 Manresa 08242 Spain

Many bodies around the world make their decisions through voting systems in which voters have several options and the collective result also has several options. Many of these voting systems are anonymous, i.e., all voters have an identical role in voting. Anonymous simple voting games, a binary vote for voters and a binary collective decision, can be represented by an easy weighted game, i.e., by means of a quota and an identical weight for the voters. Widely used voting systems of this type are the majority and the unanimity decision rules. In this article, we analyze the case in which voters have two or more voting options and the collective result of the vote has also two or more options. We prove that anonymity implies being representable through a weighted game if and only if the voting options for voters are binary. As a consequence of this result, several significant enumerations are obtained.

关键词： Decision-making Multichoice games Anonymous decision systems Weighted decision systems pseudo-boolean functions Enumerations

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Do additional target points speed up evolutionary algorithms

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THEORETICAL COMPUTER SCIENCE 2023年 950卷

作者： Bossek, Jakob Sudholt, Dirk Rhein Westfal TH Aachen Depatment Comp Sci Aachen Germany Univ Passau Germany Fuculty Comp Sci & Math Passau Germany

Most runtime analyses of randomised search heuristics .ocus on the expected number of function valuations to find a unique global optimum. We ask a fundame..tal question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPUS where S is a set of non-optimal search points and OPI the set of optima and compare it to the expected hitting time of OPT. We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions with polynomial expected optimisation times we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering *** balls around all global optima gives an easier target for some algorithm and functions and can shift the phase transition with respect to offspring population sites in the (1,A) EA on ONE! *AX. However, for the one-dimensional Ising model the time to reach Hamming halls of radius (1/2) around optima does not reduce the asymptotic expected optimisation time in the worst case. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time. (c) 2023 Elsevier B.V. All rights reserved.

关键词： Evolution2ry algorithms pseudo-boolean functions Runtime analysis

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Do Additional Optima Speed Up Evolutionary Algorithms? 21

Do Additional Optima Speed Up Evolutionary Algorithms?

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16th ACM/SIGEVO Conference on Foundations of Genetic Algorithms (FOGA)

作者： Bossek, Jakob Sudholt, Dirk Univ Munster Dept Informat Syst Munster Germany Univ Passau Dept Comp Sci & Math Passau Germany

ISBN: (纸本)9781450383523

Most runtime analyses of randomised search heuristics focus on the expected number of function evaluations to find a unique global optimum. We ask a fundamental question: if additional search points are declared optimal, or declared as desirable target points, do these additional optima speed up evolutionary algorithms? More formally, we analyse the expected hitting time of a target set OPT u S where S is a set of non-optimal search points and OPT is the set of optima and compare it to the expected hitting time of OPT. We show that the answer to our question depends on the number and placement of search points in S. For all black-box algorithms and all fitness functions we show that, if additional optima are placed randomly, even an exponential number of optima has a negligible effect on the expected optimisation time. Considering Hamming balls around all global optima gives an easier target for some algorithms and functions and can shift the phase transition with respect to offspring population sizes in the (1,lambda) EA on ONE-MAX. Finally, on functions where search trajectories typically join in a single search point, turning one search point into an optimum drastically reduces the expected optimisation time.

关键词： Evolutionary algorithms pseudo-boolean functions runtime analysis theory

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An initial step toward a quantum annealing approach to the discrete logarithm problem

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SECURITY AND PRIVACY 2022年第4期5卷

作者： Mahasinghe, Anuradha Jayasinghe, Youvin Univ Colombo Dept Math Ctr Math Modeling Colombo Sri Lanka

The discrete logarithm problem over a multiplicative group of integer modulo n is the key ingredient in the ElGamal encryption system. This problem has been proven to be tractable in polynomial time on a quantum computer by Shor's algorithm. This algorithm makes use of basic quantum gates, circuits, measurements and it has been experimented in the circuit-gate framework of quantum computing. On the other hand, the adiabatic framework of quantum computing with the quantum annealers such as the D-Wave machine that simulates the adiabatic process has become very popular, showing its potential for making a practical pathway to achieve quantum speedup. Furthermore, much progress has been reported recently on the tractability of the integer factoring problem, which is the other widely-used encryption method, on quantum annealers. In this context, it is important to explore the tractability of the discrete logarithm problem too using quantum annealers. In this work we present a first step made in this direction, by converting the discrete logarithm problem over a multiplicative group into the problem of minimizing a binary quadratic form, a standard problem format acceptable to a quantum annealer. Our formulation was tested for small-scale problem instances using the quantum-classical hybrid platform PyQUBO and we discuss the computational challenges and propose several potential improvements to the formulation.

关键词： cryptanalysis cryptosystems optimization pseudo-boolean functions quadratic forms quantum annealing quantum computing

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