pseudo-Boolean Optimization Problems Research Articles

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.

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.

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.

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.

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.

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.

A BRANCH-AND-BOUND ALGORITHM FOR A PSEUDO-BOOLEAN OPTIMIZATION PROBLEM WITH BLACK-BOX FUNCTIONS

We consider a conditional pseudo-Boolean optimization problem with both the objective function and all constraint functions given algorithmically (black-box functions) and defined on {0, 1}n only. We suppose that these functions have certain properties, for example, unimodality and monotonicity. To solve problems of this type, we propose an optimization algorithm based on finding boundary points of the feasible region and the branch-and-bound method. The developed algorithm is aimed at the reception of an exact solution of an optimization problem. In addition, this algorithm can be used as an improvement of approximate algorithms such as the greedy heuristic and the random search algorithms for finding boundary points. Even after a small number of iterations (branchings), a significant improvement of the found feasible solution is achieved.

Boolean-Based Dependency Management for the Eclipse Ecosystem

In June 2008, the Eclipse open platform released a new dependency management system called p2. That system was based on the translation of the dependency management problem into a pseudo-Boolean optimization problem, to be handled by the Sat4j solver. Since then, p2 has been more tightly integrated with Sat4j, the platform opened a public plugin repository (the Eclipse marketplace) which relies on p2 to install the available plugins and their dependencies, and became the favorite way to install plugins in the Eclipse community. This paper summarizes the issues raised by Eclipse dependency management, its pseudo-Boolean encoding within p2, its extension for Linux package management with p2cudf, and concludes with lessons learned on using research software in production systems.

Drift analysis of ant colony optimization of stochastic linear pseudo-boolean functions

In this paper we study the behavior of a variant of the Max–Min Ant System algorithm when applied to a stochastic Linear Pseudo-Boolean Optimization problem. Previous related work is on a partial analysis of its performance on a different problem. Here, we carry out its complete performance analysis giving a bound on its average runtime using drift analysis. For the purpose, we give a new drift theorem and use it to analyze the algorithm when applied to our problem.

Gray Box Optimization for Mk Landscapes (NK Landscapes and MAX-kSAT).

This article investigates Gray Box Optimization for pseudo-Boolean optimization problems composed of M subfunctions, where each subfunction accepts at most k variables. We will refer to these as Mk Landscapes. In Gray Box Optimization, the optimizer is given access to the set of M subfunctions. We prove Gray Box Optimization can efficiently compute hyperplane averages to solve non-deceptive problems in [Formula: see text] time. Bounded separable problems are also solved in [Formula: see text] time. As a result, Gray Box Optimization is able to solve many commonly used problems from the evolutional computation literature in [Formula: see text] evaluations. We also introduce a more general class of Mk Landscapes that can be solved using dynamic programming and discuss properties of these functions. For certain type of problems Gray Box Optimization makes it possible to enumerate all local optima faster than brute force methods. We also provide evidence that randomly generated test problems are far less structured than those found in real-world problems.

A Multilevel Pseudo-Boolean Satisfiability-Based Approach for Segmented Channel Routing

The segmented channel routing problem is fundamental to the routing of row-based field programmable gate arrays (FPGAs) and is proven to be nondeterministic polynomial time (NP) complete. In this paper, we capitalize on the compelling advancements in satisfiability (SAT) solvers to propose a multilevel pseudo-Boolean SAT-based approach. We construct several levels of hierarchy amongst the nets and the routing problem of each level is formulated as a pseudo-Boolean optimization (PBO) problem. Moreover, an optimization technique of reducing the number of variables in PBO problems is described to improve the scalability of the proposed method. Similar to the SAT-based routing, the unroutability of a given circuit can be proved by the approach. Experimental results show that the proposed method compares very favorably with existing algorithms and achieves the best convergence rate.

Markov random field based phase demodulation of interferometric images

We present a novel method to solve the sign ambiguity for phase demodulation from a single interferometric image that possibly contains closed fringes. The problem is formulated in a Markov random field (MRF) energy minimization framework with the assumption of phase gradient orientation continuity. The binary pairwise objective function is non-submodular and therefore its minimization is an NP-hard problem, for which we devise a multigrid hierarchy of quadratic pseudoboolean optimization problems that can be improved iteratively to approximate the optimal solution. We name the method MSARI algorithm, for Markov based sign ambiguity resolution in interferometry. Compared with traditional path-following phase demodulation methods, the new approach does not require any heuristic scanning strategy, is not subject to the propagation of error, and the extension to three dimensional fringe patterns is straightforward. A set of experiments with synthetic data and real prelens tear film interferometric images of the human eye demonstrate the effectiveness and robustness of the proposed algorithm as compared with existing state-of-the-art phase demodulation methods.

Using DRL* relaxations for quadratically constrained pseudoboolean optimization: application to robust Min-Cut

In this work we focus on solving quadratically constrained pseudoboolean optimization problems with quadratic objective as mixed integer linear programs. The standard mixed integer linear formulation of such problems is strengthened using valid inequalities derived from solving Reformulation-Linearization relaxation called partial DRL* relaxation. The proposed PDRL* relaxation features block-decomposable structure which are exploited to improve computational efficiency. We present computational results obtained with the rank 2 PDRL*, showing that the proposed mixed integer linear formulation gives rise to significant reduction factors (typically more than 1000) in the size of the branch and bound trees on instances of robust minimum cut problem with weight constraints.

Comparison of PBO solvers in a dependency solving domain

Linux package managers have to deal with dependencies and conflicts of packages required to be installed by the user. As an NP-complete problem, this is a hard task to solve. In this context, several approaches have been pursued. Apt-pbo is a package manager based on the apt project that encodes the dependency solving problem as a pseudo-Boolean optimization (PBO) problem. This paper compares different PBO solvers and their effectiveness on solving the dependency solving problem.

DFT and Minimum Leakage Pattern Generation for Static Power Reduction During Test and Burn-In

This paper presents a design for testability and minimum leakage pattern generation technique to reduce static power during test and burn-in for nanometer technologies. This technique transforms the minimum leakage pattern generation problem into a pseudo-Boolean optimization (PBO) problem. Nonlinear objective functions of leakage power are approximated by linear ones such that this problem can be solved efficiently by an existing PBO solver. A partitioning-based algorithm is applied for control point insertion and also CPU time reduction. Experimental results on the IEEE ISCAS'89 benchmark circuits using Taiwan Semiconductor Manufacturing Company 90-nm technology show that, for large circuits, the static power is reduced from 8.3% (without partition) to 17.47% (with 64 partitions). Besides, the overall CPU time is reduced from 3600 s (without partition) to 83 s (with 64 partitions). This technique reduces the static power without changing the manufacturing process or library cells.

A UNIFYING FRAMEWORK FOR GENERALIZED CONSTRAINT ACQUISITION

When a practical problem can be modeled as a constraint satisfaction problem (CSP), which is a set of constraints that need to be satisfied, it can be solved using many constraint programming techniques. In many practical applications, while users can recognize examples of where a CSP should be satisfied or violated, they cannot articulate the specification of the CSP itself. In these situations, it can be helpful if the computer can take an active role in learning the CSP from examples of its solutions and non-solutions. This is called constraint acquisition. This paper introduces a framework for constraint acquisition in which one can uniformly define and formulate constraint acquisition problems of different types as optimization problems. The difference between constraint acquisition problems within the framework is not only in the type of constraints that need to be acquired but also in the learning objective. The generic framework can be instantiated to obtain specific formulations for acquiring classical, fuzzy, weighted or probabilistic constraints. The paper shows as an example how recent techniques for acquiring classical constraints can be directly obtained from the framework. Specifically, the formulation obtained from the framework to acquire classical CSPs with the minimum number of violated examples is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable by using many available optimization tools. The paper also reports empirical results on constraint acquisition methods to show the utility of the framework.

Synthesis of Narrowing Enquiries

A canonical model of decision making, with a disjunctive restriction [1] and the concept of a Pareto set Π(A) which describes the uncertainty region [3], is used as a basis for developing a method for active narrowing enquiries for solving canonical pseudoboolean optimization problems.

An Approximation Technique for Pseudo-Boolean Maximization Problems

Abstract A partitioning procedure is proposed for obtaining approximate solutions to pseudo-Boolean optimization problems (e.g., zero-one polynomial programs). Such a procedure is useful not only in its ability to provide a bound value but further as a means for initializing optimum seeking methods, such as implicit enumeration techniques, employed to insure optimality.