pseudo-Boolean Optimization Research Articles

Runtime Analysis of Quality Diversity Algorithms

AbstractQuality diversity (QD) is a branch of evolutionary computation that gained increasing interest in recent years. The Map-Elites QD approach defines a feature space, i.e., a partition of the search space, and stores the best solution for each cell of this space. We study a simple QD algorithm in the context of pseudo-Boolean optimisation on the “number of ones” feature space, where the ith cell stores the best solution amongst those with a number of ones in $$[(i-1)k, ik-1]$$ [ ( i - 1 ) k , i k - 1 ] . Here k is a granularity parameter $$1 \le k \le n+1$$ 1 ≤ k ≤ n + 1 . We give a tight bound on the expected time until all cells are covered for arbitrary fitness functions and for all k and analyse the expected optimisation time of QD on OneMax and other problems whose structure aligns favourably with the feature space. On combinatorial problems we show that QD finds a $${(1-1/e)}$$ ( 1 - 1 / e ) -approximation when maximising any monotone sub-modular function with a single uniform cardinality constraint efficiently. Defining the feature space as the number of connected components of an edge-weighted graph, we show that QD finds a minimum spanning forest in expected polynomial time. We further consider QD’s performance on classes of transformed functions in which the feature space is not well aligned with the problem. The asymptotic performance is unaffected by transformations on easy functions like OneMax. Applying a worst-case transformation to a deceptive problem increases the expected optimisation time from $$O(n^2 \log n)$$ O ( n 2 log n ) to an exponential time. However, QD is still faster than a (1+1) EA by an exponential factor.

Large-scale and cooperative graybox parallel optimization on the supercomputer Fugaku

We design, develop and analyze parallel variants of a state-of-the-art graybox optimization algorithm, namely Drils (Deterministic Recombination and Iterated Local Search), for attacking large-scale pseudo-boolean optimization problems on top of the large-scale computing facilities offered by the supercomputer Fugaku. We first adopt a Master/Worker design coupled with a fully distributed Island-based model, ending up with a number of hybrid OpenMP/MPI implementations of high-level parallel Drils versions. We show that such a design, although effective, can be substantially improved by enabling a more focused iteration-level cooperation mechanism between the core graybox components of the original serial Drils algorithm. Extensive experiments are conducted in order to provide a systematic analysis of the impact of the designed parallel algorithms on search behavior, and their ability to compute high-quality solutions using increasing number of CPU-cores. Results using up to 1024×12-cores NUMA nodes, and NK-landscapes with up to 10,000 binary variables are reported, providing evidence on the relative strength of the designed hybrid cooperative graybox parallel search.

Estimation-of-distribution algorithms for multi-valued decision variables

The majority of research on estimation-of-distribution algorithms (EDAs) concentrates on pseudo-Boolean optimization and permutation problems, leaving the domain of EDAs for problems in which the decision variables can take more than two values, but which are not permutation problems, mostly unexplored. To render this domain more accessible, we propose a natural way to extend the known univariate EDAs to this setting. Different from a naïve reduction to the binary case, our approach avoids additional constraints.Since understanding genetic drift is crucial for an optimal parameter choice, we extend the known quantitative analysis of genetic drift to EDAs for multi-valued, categorical variables. Roughly speaking, when the variables take r different values, the time for genetic drift to become significant is r times shorter than in the binary case. Consequently, the update strength of the probabilistic model has to be chosen r times lower now.To investigate how desired model updates take place in this framework, we undertake a mathematical runtime analysis on the r-valued LeadingOnes problem. We prove that with the right parameters, the multi-valued UMDA solves this problem efficiently in O(rln⁡(r)2n2ln⁡(n)) function evaluations. This bound is nearly tight as our lower bound Ω(rln⁡(r)n2ln⁡(n)) shows.Overall, our work shows that our good understanding of binary EDAs naturally extends to the multi-valued setting, and it gives advice on how to set the main parameters of multi-values EDAs.

Iterated Local Search with Linkage Learning

In pseudo-Boolean optimization, a variable interaction graph represents variables as vertices, and interactions between pairs of variables as edges. In black-box optimization, the variable interaction graph may be at least partially discovered by using empirical linkage learning techniques. These methods never report false variable interactions, but they are computationally expensive. The recently proposed local search with linkage learning discovers the partial variable interaction graph as a side-effect of iterated local search. However, information about the strength of the interactions is not learned by the algorithm. We propose local search with linkage learning 2, which builds a weighted variable interaction graph that stores information about the strength of the interaction between variables. The weighted variable interaction graph can provide new insights about the optimization problem and behavior of optimizers. Experiments with NK landscapes, knapsack problem, and feature selection show that local search with linkage learning 2 is able to efficiently build weighted variable interaction graphs. In particular, experiments with feature selection show that the weighted variable interaction graphs can be used for visualizing the feature interactions in machine learning. Additionally, new transformation operators that exploit the interactions between variables can be designed. We illustrate this ability by proposing a new perturbation operator for iterated local search.

Run Time Analysis for Random Local Search on Generalized Majority Functions

Run time analysis of evolutionary algorithms recently makes significant progress in linking algorithm performance to algorithm parameters. However, settings that study the impact of problem parameters are rare. The recently proposed W-model provides a good framework for such analyses, generating pseudo-Boolean optimization problems with tunable properties. We initiate theoretical research of the W-model by studying how one of its properties-neutrality-influences the run time of random local search. Neutrality creates plateaus in the search space by first performing a majority vote for subsets of the solution candidate and then evaluating the smaller-dimensional string via a low-level fitness function. We prove upper bounds for the expected run time of random local search on this MAJORITY problem for its entire parameter spectrum. To this end, we provide a theorem, applicable to many optimization algorithms, that links the run time of MAJORITY with its symmetric version HASMAJORITY, where a sufficient majority is needed to optimize the subset. We also introduce a generalized version of classic drift theorems as well as a generalized version of Wald’s equation, both of which we believe to be of independent interest.

Fast 1-flip neighborhood evaluations for large-scale pseudo-Boolean optimization using posiform representation

We consider the general case of pseudo-Boolean optimization (PBO). This problem belongs to the NP-hard class of computational complexity and generalizes the well-known quadratic unconstrained binary optimization (QUBO) problem, which is able to model a wide range of combinatorial optimization problems and also has applications in quantum computing. Several state-of-the-art methods in the literature for solving specific classes of PBO problems rely on the ability to quickly evaluate changes in objective function value upon a flip move, i.e., after changing the value of a Boolean variable in a given solution from 0 to 1 or from 1 to 0. In this work, we propose closed-form formulae, and develop algorithms and auxiliary data structures for quickly evaluating 1-flip move neighborhoods in PBO problems. We work with the objective function represented as a posiform, i.e., a sum of terms that may contain Boolean variables or negations thereof. This representation is interesting for solving large-scale instances, since converting a posiform to an expression containing no variable negations may take exponential time and space in the number of variables. We also provide implementations of our algorithms and data structures as a library written in C programming language. This library can be used to implement methods with fast flip move evaluations for solving PBO problem instances. We conducted computational experiments with implementations of a Iterated Tabu Search (ITS) metaheuristic, when solving randomly-generated instances, and also instances of the Hypergraph Maximum Cut problem, which can be easily recast as a PBO problem. The results showed that our best evaluation algorithms provided relatively large speedups as instance sizes increased, and became more competitive as a consequence of speedup.

APPROACH TO SOLVING THE PROBLEM OF SECTOR FORMATION
 IN ORGANIZING THE WORK OF DRIVERS
 AT THE ENTERPRISES OF ROUTE PASSENGER TRANSPORT
 BASED ON PSEUDOBOOLEAN OPTIMIZATION

An approach to solving the optimization problem arising during formation of sectors at the enterprises of route passenger transport by the sectoral method is considered, which allows to increase the efficiency of the mode of using drivers' working hours and reduce unproductive costs of the enterprise. The presented approach and its implementation based on pseudo-boolean optimization with the use of smooth optimization methods, in particular, “inner point” method have found their application in organization of drivers' work.

Automated Configuration of Genetic Algorithms by Tuning for Anytime Performance

Finding the best configuration of algorithms’ hyperparameters for a given optimization problem is an important task in evolutionary computation. We compare in this work the results of four different hyperparameter optimization (HPO) approaches for a family of genetic algorithms (GAs) on 25 diverse pseudo-Boolean optimization (PBO) problems. More precisely, we compare previously obtained results from a grid search with those obtained from three automated configuration techniques: 1) iterated racing; 2) mixed-integer parallel-efficient global optimization (MIP-EGO); and 3) mixed-integer evolutionary strategies. Using two different cost metrics: 1) expected running time (ERT) and 2) the area under the empirical cumulative distribution function (ECDF) curve, we find that in several cases the best configurations with respect to ERT are obtained when using the area under the ECDF curve as the cost metric during the configuration process. Our results suggest that even when interested in ERT performance, it might be preferable to use anytime performance measures for the configuration task. We also observe that tuning for ERT is much more sensitive with respect to the budget that is allocated to the target algorithms.

Pseudo-Boolean optimisation for RobinX sports timetabling

We report on the development of Reprobate, a tool for solving sports timetabling problems in a subset of the RobinX format. Our tool is based around a monolithic translation of a sports timetabling instance into a pseudo-Boolean (PB) optimisation problem; this instance can be solved using existing pseudo-Boolean solvers. Once the tool has found a feasible solution, it can improve it using a second encoding that alters only the home/away pattern of games. We entered our tool into the International Timetabling Competition 2021. While it was effective on many instances, it struggled to cope with schedules involving large break constraints. However, among instances for which it could initially find a feasible solution, the combination of use of a portfolio of solvers, a range of variations on the encoding and the aforementioned local improvement process yielded an average reduction in solution cost of 23%.

Power-aware test scheduling framework for IEEE 1687 multi-power domain networks using formal techniques

The IEEE 1687 Std. (IJTAG) introduces an efficient access methodology based on reconfigurable scan networks to address the ever-increasing complexity of the latest system-on-chips. By invoking this new methodology, the overall test time can considerably be reduced by shortening the scan chains' length without sacrificing the test quality. IJTAG allows for designing highly complex test networks compromising the latest test structure that enables to achieve the high test quality, as required by today's customers' applications. Besides the time overhead induced by the required network (re-)configuration, the access sequence of the instruments itself greatly affects the overall test time. Furthermore, the power characteristics of the complex test facilities have to be taken into account to avoid effects like IR drop during the later test application that is even more critical in highly multi-power domain networks. Consequently, the IJTAG methodology strictly requires an effective test scheduler that considers the individual instruments' constraints and is compatible with recent multi-power domain chip designs. This work proposes two novel power-aware test scheduling approaches based on pseudo-Boolean optimization and integer linear programming techniques that are both seamlessly integrated into a fully-automated test scheduling framework for IJTAG test networks even with multiple power domains. The first proposed optimization scheme allows for determining a local optimal test schedule, applicable to networks with more than one thousand instruments since the required run-time is manageable. Furthermore, the second optimization scheme determines a global optimal test scheduler, which is most suitable for mid-sized networks in which it is clearly outperforming any other existing technique.

Quantum-Inspired Hierarchy for Rank-Constrained Optimization

Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs. Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.

Fast 1-Flip Neighborhood Evaluations for Large-Scale Pseudo-Boolean Optimization Using Posiform Representation

Fast 1-Flip Neighborhood Evaluations for Large-Scale Pseudo-Boolean Optimization Using Posiform Representation

Pseudo-Boolean Conditional Optimization Models for a Class of Multiple Traveling Salesmen Problems

Pseudo-Boolean Conditional Optimization Models for a Class of Multiple Traveling Salesmen Problems

Cutting to the Core of Pseudo-Boolean Optimization: Combining Core-Guided Search with Cutting Planes Reasoning

Core-guided techniques have revolutionized Boolean satisfiability approaches to optimization problems (MaxSAT), but the process at the heart of these methods, strengthening bounds on solutions by repeatedly adding cardinality constraints, remains a bottleneck. Cardinality constraints require significant work to be re-encoded to SAT, and SAT solvers are notoriously weak at cardinality reasoning. In this work, we lift core-guided search to pseudo-Boolean (PB) solvers, which deal with more general PB optimization problems and operate natively with cardinality constraints. The cutting planes method used in such solvers allows us to derive stronger cardinality constraints, which yield better updates to solution bounds, and the increased efficiency of objective function reformulation also makes it feasible to switch repeatedly between lower-bounding and upper- bounding search. A thorough evaluation on applied and crafted benchmarks shows that our core-guided PB solver significantly improves on the state of the art in pseudo-Boolean optimization.

Algorithm for assortment planning based on the method of changing probabilities

The paper considers the problem that manufacturing enterprises with complex diversified production are faced with a number of specific problems related to the formation of their range of products (goods). For these problems solving such organizations (sales department, marketing department, commercial service) create a certain service that deals with solving a set of tasks, such as studying demand, searching for goods from suppliers, managing selling prices, determining optimal volumes of production or purchasing goods, managing the assortment of an enterprise, coordinating the actions of other services related to the promotion of goods. The solution of local problems (tasks of operational management) at the level of a separate direction, project, contract, a separate batch of goods due to the extreme variety of products under consideration, puts forward some additional informal requirements and criteria for evaluating the effectiveness of solutions, which, due to their diversity and complexity of the strict mathematical description, remain beyond those areas of management theory that operate on formalized data and criteria. Nevertheless, the development of the competitive environment, the tightening requirements for the efficiency of the production use, financial, information assets leads to the impossibility to estimate all possible solutions qualitatively (adequately) and choose a really best alternative (or one of the best solutions). It helps to describe the behaviour of the sales market according to separate goods or product groups taking into account the peculiarities of the consumer’s attitude to a certain set of nomenclature based mainly on objective data reflected in operational records. The paper proposes an algorithm for automating business procedures for preparing draft of decisions on the purchase of batches of goods (пaртий тoвaрoв) in the form of a pseudo-Boolean optimization problem.

Towards a Better Basis Search through a Surrogate Model-Based Epistasis Minimization for Pseudo-Boolean Optimization

Epistasis, which indicates the difficulty of a problem, can be used to evaluate the basis of the space in which the problem lies. However, calculating epistasis may be challenging as it requires all solutions to be searched. In this study, a method for constructing a surrogate model, based on deep neural networks, that estimates epistasis is proposed for basis evaluation. The proposed method is applied to the Variant-OneMax problem and the NK-landscape problem. The method is able to make successful estimations on a similar level to basis evaluation based on actual epistasis, while significantly reducing the computation time. In addition, when compared to the epistasis-based basis evaluation, the proposed method is found to be more efficient.

A semantic relatedness preserved subset extraction method for language corpora based on pseudo-Boolean optimization

As language corpora have been playing an increasingly important role in the field of Artificial Intelligence (AI) research, lots of extremely large corpora are created. However, a larger corpora size not only increases power and accuracy but also brings redundancy. Therefore, researchers began to emphasize the study of appropriate subset extraction methods. Due to the trade-off between data sufficiency and redundancy, a group of interesting and challenging problems are emerged that are studied in this paper: (1) How to make the resulting subset include as much data as possible under some necessary constraints? (2) How to preserve the potential useful semantic relatedness included in the original corpora while reducing the size of the corpora? For these two problems, existing work mainly focuses on the methods to construct particular subsets for special usage. These methods are limited in their focus. In this paper, we try to address the problems listed above. First, considering the cubic and binary semantic relatedness among tokens, we construct a general system model and formulate the mix problem as a cubic pseudo-Boolean optimization problem. Then, by analyzing the characteristics of the objective function, we transfer the problem into the maximum flow problem of a corresponding graph. Third, we propose a new algorithm by introducing discrete Lagrangian iteration method. We prove that the objective function is supermodular, which allows us to use fast minimum cut algorithms in each iteration step to propose another fast algorithm. Finally, we experimentally validate our new algorithms on several randomly created corpora.

Finding Most Compatible Phylogenetic Trees over Multi-State Characters

The reconstruction of the evolutionary tree of a set of species based on qualitative attributes is a central problem in phylogenetics. In the NP-hard perfect phylogeny problem the input is a set of taxa (species) and characters (attributes) on them, and the task is to find an evolutionary tree that describes the evolution of the taxa so that each character state evolves only once. However, in practical situations a perfect phylogeny rarely exists, motivating the maximum compatibility problem of finding the largest subset of characters admitting a perfect phylogeny. Various declarative approaches, based on applying integer programming (IP), answer set programming (ASP) and pseudo-Boolean optimization (PBO) solvers, have been proposed for maximum compatibility. In this work we develop a new hybrid approach to solving maximum compatibility for multi-state characters, making use of both declarative optimization techniques (specifically maximum satisfiability, MaxSAT) and an adaptation of the Bouchitt'e-Todinca approach to triangulation-based graph optimization problems. Empirically our approach outperforms in scalability the earlier proposed approaches w.r.t. various parameters underlying the problem.

SAT-Based Generation of Optimum Circuits with Polymorphic Behavior Support

This paper presents a method for generating optimum multi-level implementations of Boolean functions based on Satisfiability (SAT) and Pseudo-Boolean Optimization (PBO) problems solving. The method is able to generate one or enumerate all optimum implementations, while the allowed target gate types and gates costs can be arbitrarily specified. Polymorphic circuits represent a newly emerging computation paradigm, where one hardware structure is capable of performing two or more different intended functions, depending on instantaneous conditions in the target operating environment. In this paper we propose the first method ever, generating provably size-optimal polymorphic circuits. Scalability and feasibility of the method are documented by providing experimental results for all NPN-equivalence classes of four-input functions implemented in AND–Inverter and AND–XOR–Inverter logics without polymorphic behavior support being used and for all pairs of NPN–equivalence classes of three-input functions for polymorphic circuits. Finally, several smaller benchmark circuits were synthesized optimally, both in standard and polymorphic logics.

Bayesian partition crossover for pseudo-Boolean optimization

Bayesian partition crossover for pseudo-Boolean optimization