Range Of Combinatorial Optimization Problems Research Articles

Counting the number of solutions in satisfiability problems with tensor-network message passing.

We investigate how to count the number of solutions in the Boolean satisfiability (SAT) problem, a fundamental problem in theoretical computer science that has applications in various domains. We convert the problem to a spin-glass problem in statistical physics and approximately compute the entropy of the problem, which corresponds to the logarithm of the number of solutions. We propose a new method for the entropy computing problem based on a combination of the tensor network method and the message-passing algorithm. The significance of the proposed method is its ability to consider the effects of both long loops and short loops present in the factor graph of the SAT problem. We validate the efficacy of our approach using 3-SAT problems defined on the random graphs and structured graphs, and show that the proposed method gives more accurate results on the number of solutions than the standard belief propagation algorithms. We also discuss the applications of our method across a wide range of combinatorial optimization problems.

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.

Quantum Optimization with Arbitrary Connectivity Using Rydberg Atom Arrays

Programmable quantum systems based on Rydberg atom arrays have recently been used for hardware-efficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits. In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system. Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from a wide class of problems to maximum-weighted independent set problems on unit-disk graphs, with at most a quadratic overhead in the number of qubits. We analyze several examples, including maximum-weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization. Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems. Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry.5 MoreReceived 14 September 2022Revised 14 December 2022Accepted 4 January 2023DOI:https://doi.org/10.1103/PRXQuantum.4.010316Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBoolean satisfiability problemNP-hard problemsQuantum information architectures & platformsQuantum softwarePhysical SystemsAtomsRydberg atoms & moleculesQuantum InformationAtomic, Molecular & OpticalStatistical Physics

Negative Learning Ant Colony Optimization for MaxSAT

Recently, a new negative learning variant of ant colony optimization (ACO) has been used to successfully tackle a range of combinatorial optimization problems. For providing stronger evidence of the general applicability of negative learning ACO, we investigate how it can be adapted to solve the Maximum Satisfiability problem (MaxSAT). The structure of MaxSAT is different from the problems considered to date and there exists only a few ACO approaches for MaxSAT. In this paper, we describe three negative learning ACO variants. They differ in the way in which sub-instances are solved at each algorithm iteration to provide negative feedback to the main ACO algorithm. In addition to using IBM ILOG CPLEX, two of these variants use existing MaxSAT solvers for this purpose. The experimental results show that the proposed negative learning ACO variants significantly outperform the baseline ACO as well as IBM ILOG CPLEX and the two MaxSAT solvers. This result is of special interest because it shows that negative learning ACO can be used to improve over the results of existing solvers by internally using them to solve smaller sub-instances.

A Self-Adaptive Variant of CMSA: Application to the Minimum Positive Influence Dominating Set Problem

Construct, merge, solve and adapt (CMSA) is a recently developed, generic algorithm for combinatorial optimisation. Even though the usefulness of the algorithm has been demonstrated by applications to a range of combinatorial optimisation problems, in some applications, it was observed that the algorithm can be sensitive to parameter settings. In this work, we propose a self-adaptive variant of CMSA, called Adapt-CMSA, with the aim of reducing the parameter sensitivity of the original version of CMSA. The advantages of this new CMSA variant are demonstrated in the context of the application to the so-called minimum positive influence dominating set problem. It is shown that, in contrast to CMSA, Adapt-CMSA does not require a computation time intensive parameter tuning process for subsets of the considered set of problem instances. In fact, after tuning Adapt-CMSA only once for the whole set of benchmark instances, the algorithm already obtains state-of-the-art results. Nevertheless, note that the main objective of this paper is not the tackled problem but the improvement of CMSA.

GPU-accelerated scalable solver with bit permutated cyclic-min algorithm for quadratic unconstrained binary optimization

A wide range of combinatorial optimization problems can be reduced to the Ising model, and equivalently the quadratic unconstrained binary optimization (QUBO) problem. Thus, in recent years, researchers have proposed to solve QUBO on FPGAs, GPUs, and special-purpose processors. The adaptive bulk search (ABS) is a previously-proposed framework for solving QUBO in parallel on multiple GPUs. In the ABS, a CPU host performs a GA-based global search while GPUs asynchronously perform many local searches in parallel. An original ABS adopts a simple local search algorithm called a cyclic-min algorithm, which does not use pseudo random numbers. However, the lack of randomness may cause a potential drawback of restricted bit-flipping operations in a local search. To avoid this drawback, this paper proposes a cyclic-min algorithm with randomly-generated multiple bit permutations, which enables a more effective local search with random number generation in CPUs (not in GPUs). Furthermore, this paper introduces a scalable implementation of the ABS with MPI and OpenMP. Our experimental results on TSUBAME3.0 show that the solution quality improves and the throughput linearly increases as the number of GPUs increases; with 256 GPUs, it evaluates 20.1×1012 solutions per second.

A Probabilistic Compute Fabric Based on Coupled Ring Oscillators for Solving Combinatorial Optimization Problems

Nondeterministic polynomial time hard (NP-hard) combinatorial optimization problems (COPs) are intractable to solve using a traditional computer as the time to find a solution increases very rapidly with the number of variables. An efficient alternative computing method uses coupled spin networks to solve COP. This work presents a first-of-its-kind coupled ring oscillator (ROSC)-based scalable probabilistic Ising computer to solve NP-hard COPs. An integrated coupled oscillator network was designed with 560 ROSCs that mimic a coupled spin network. Each ROSC can be coupled to any of its neighbors using programmable back-to-back (B2B) inverter-based coupling mechanism. The ROSC-based spins and B2B inverter-based coupling were optimized to work under a wide range of system noise as well as voltage and temperature variations. Randomly generated 1000 max-cut problems were mapped and solved in the hardware. The integrated Ising computer produced satisfactory solutions of max-cut problems when compared with commercial software running on a CPU. Experiments show that the integrated CMOS-based Ising computer can find the solution to NP-hard problems with an accuracy of 82%-100%. In addition, the repeated measurements of the same problem showed that the Ising computer can traverse through several local minima to find high-quality solutions under various voltage and temperature variation conditions. The experimental results show that ROSCs are a potential candidate for a dedicated hardware accelerator aiming to solve a wide range of COPs.

Parallel transfer evolution algorithm

Parallelization of an evolutionary algorithm takes the advantage of modular population division and information exchange among multiple processors. However, existing parallel evolutionary algorithms are rather ad hoc and lack a capability of adapting to diverse problems. To accommodate a wider range of problems and to reduce algorithm design costs, this paper develops a parallel transfer evolution algorithm. It is based on the island-model of parallel evolutionary algorithm and, for improving performance, transfers both the connections and the evolutionary operators from one sub-population pair to another adaptively. Needing no extra upper selection strategy, each sub-population is able to select autonomously evolutionary operators and local search operators as subroutines according to both the sub-population’s own and the connected neighbor’s ranking boards. The parallel transfer evolution is tested on two typical combinatorial optimization problems in comparison with six existing ad-hoc evolutionary algorithms, and is also applied to a real-world case study in comparison with five typical parallel evolutionary algorithms. The tests show that the proposed scheme and the resultant PEA offer high flexibility in dealing with a wider range of combinatorial optimization problems without algorithmic modification or redesign. Both the topological transfer and the algorithmic transfer are seen applicable not only to combinatorial optimization problems, but also to non-permutated complex problems.

Machine Learning Approaches to Learning Heuristics for Combinatorial Optimization Problems

This paper examines various ways of utilizing machine learning to enhance or develop heuristic and metaheuristic algorithms for combinatorial optimization problems (COPs). A general agent-based framework is introduced, which enables learning both constructive and perturbative heuristics for a wide range of COPs. The framework consists of an agent-based generalized solution algorithm (GSA) to COPs, along with a machine learning architecture and training procedure that would inject intelligence via learning to the agents. A GSA in our framework is an assembly of k agents working in parallel, generating the solution to a given problem one step at a time, either by improving a known solution or by constructing one from scratch. The generated partial solution is then implemented in a simulated or real system that represents the environment, with state changes being fed back to the k agents. In the proposed framework, intelligence can be injected into the agents individually or in groups, via learning from the correlation of their solution decisions with a final objective value. A discussion is also provided on what machine learning approaches to use depending on the availability of training data, which in turn depends on the choice of algorithm architecture.

Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis

The Quadratic Unconstrained Binary Optimization problem (QUBO) has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this article, we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits. We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(2), 79–97 2017

Guided genetic algorithm for the multidimensional knapsack problem

Genetic Algorithm (GA) has emerged as a powerful method for solving a wide range of combinatorial optimisation problems in many fields. This paper presents a hybrid heuristic approach named Guided Genetic Algorithm (GGA) for solving the Multidimensional Knapsack Problem (MKP). GGA is a two-step memetic algorithm composed of a data pre-analysis and a modified GA. The pre-analysis of the problem data is performed using an efficiency-based method to extract useful information. This prior knowledge is integrated as a guide in a GA at two stages: to generate the initial population and to evaluate the produced offspring by the fitness function. Extensive experimentation was carried out to examine GGA on the MKP. The main GGA parameters were tuned and a comparative study with other methods was conducted on well-known MKP data. The real impact of GGA was checked by a statistical analysis using ANOVA, t-test and Welch’s t-test. The obtained results showed that the proposed approach largely improved standard GA and was highly competitive with other optimisation methods.

Adaptive stability: general approach and examples

In this article we study the situation, where an optimal solution of a combinatorial optimization problem is known and we are equipped with the simple algorithms allowing to adapt our solution to different distortions of the initial data set. We wish to describe the distortions which allow the adaptation of the given optimal solution by the given simple algorithm preserving the optimality of the first. This approach, lying between common stability and reoptimization, is called adaptive stability. In the article we construct the conditions of adaptive stability for a range of combinatorial optimization problems having a particular structure. The obtained conditions allow to build efficiently the corresponding adaptive stability areas. The stability areas may be used for postoptimal preferring among different optimal solutions whether these are multiple solutions belonging to a single mathematical model or a set of optimal solutions corresponding to different models. The abstract concept is illustrated by two examples of combinatorial optimization problems: traveling salesman problem and task distribution problem.

Timing problems and algorithms: Time decisions for sequences of activities

Timing problems involve the choice of task execution dates within a predetermined processing sequence, and under various additional constraints or objectives such as time windows, time‐dependent costs, or flexible processing times, among others. Their efficient resolution is critical in branch and bound and neighborhood search methods for vehicle routing, project and machine scheduling, as well as in various applications in network optimization, resource allocation, and statistical inference. Timing‐related problems have been studied for years, yet research on this subject suffers from a lack of consensus, and most knowledge is scattered among operations research and applied mathematics domains. This article introduces a classification of timing problems and features, as well as a unifying multidisciplinary analysis of timing algorithms. In relation to frequent application cases within branching schemes or neighborhood searches, the efficient resolution of series of similar timing subproblems is also analyzed. A dedicated formalism of reoptimization “by concatenation” is introduced to that extent. The knowledge developed through this analysis is valuable for modeling and algorithmic design, for a wide range of combinatorial optimization problems with time characteristics, including rich vehicle routing settings and emerging nonregular scheduling applications, among others. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(2), 102–128 2015

Evolutionary algorithms and dynamic programming

Recently, it has been proven that evolutionary algorithms produce good results for a wide range of combinatorial optimization problems. Some of the considered problems are tackled by evolutionary algorithms that use a representation which enables them to construct solutions in a dynamic programming fashion. We take a general approach and relate the construction of such algorithms to the development of algorithms using dynamic programming techniques. Thereby, we give general guidelines on how to develop evolutionary algorithms that have the additional ability of carrying out dynamic programming steps. Finally, we show that for a wide class of the so-called DP-benevolent problems (which are known to admit FPTAS) there exists a fully polynomial-time randomized approximation scheme based on an evolutionary algorithm.

Natural Selection of Paths in Networks

AbstractWe present a novel algorithm that exhibits natural selection of paths in a network. If each node and weighted directed edge has a unique identifier, a path in the network is defined as an ordered list of these unique identifiers. We take a population perspective and view each path as a genotype. If each node has a node phenotype then a path phenotype is defined as the list of node phenotypes in order of traversal. We show that given appropriate path traversal, weight change and structural plasticity rules, a path is a unit of evolution because it can exhibit multiplicative growth (i.e. change it’s probability of being traversed), and have variation and heredity. Thus, a unit of evolution need not be a spatially distinct physical individual. The total set of paths in a network consists of all possible paths from the start node to a finish node. Each path phenotype is associated with a reward that determines whether the edges of that path will be multiplicatively strengthened (or weakened). A pair-wise tournament selection algorithm is implemented which compares the reward obtained by two paths. The directed edges of the winning path are strengthened, whilst the directed edges of the losing path are weakened. Edges shared by both paths are not changed (or weakened if diversity is desired). Each time a node is activated there is a probability that the path will mutate, i.e. find an alternative route that bypasses that node. This generates the potential for a novel but correlated path with a novel but correlated phenotype. By this process the more frequently traversed paths are responsible for most of the exploration. Nodes that are inactive for some period of time are lost (which is equivalent to connections to and from them being broken). This network-based natural selection compares favourably with a standard pair-wise tournament-selection based genetic algorithm on a range of combinatorial optimization problems and continuous parametric optimization problems. The network also exhibits memory of past selective environments and can store previously discovered characters for reuse in later optimization tasks. The pathway evolution algorithm has several possible implementations and permits natural selection with unlimited heredity without template replication.

An annotated bibliography of GRASP–Part II: Applications

AbstractA greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. It is a multi‐start or iterative process, in which each GRASP iteration consists of two phases, a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Since 1989, numerous papers on the basic aspects of GRASP, as well as enhancements to the basic metaheuristic, have appeared in the literature. GRASP has been applied to a wide range of combinatorial optimization problems, ranging from scheduling and routing to drawing and turbine balancing. This is the second of two papers with an annotated bibliography of the GRASP literature from 1989 to 2008. In the companion paper, algorithmic aspects of GRASP are surveyed. In this paper, we cover the literature where GRASP is applied to scheduling, routing, logic, partitioning, location, graph theory, assignment, manufacturing, transportation, telecommunications, biology and related fields, automatic drawing, power systems, and VLSI design.

An annotated bibliography of GRASP – Part I: Algorithms

AbstractA greedy randomized adaptive search procedure (GRASP) is a metaheuristic for combinatorial optimization. It is a multi‐start or iterative process, in which each GRASP iteration consists of two phases, a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Since 1989, numerous papers on the basic aspects of GRASP, as well as enhancements to the basic metaheuristic have appeared in the literature. GRASP has been applied to a wide range of combinatorial optimization problems, ranging from scheduling and routing to drawing and turbine balancing. This is the first of two papers with an annotated bibliography of the GRASP literature from 1989 to 2008. This paper covers algorithmic aspects of GRASP.

Estimation of Damage of Composite Structure using Hybrid Evolutionary Approach

Composite materials have been widely applied in a variety of engineering and industrial fields over the past two decades. In order to reduce maintenance costs and to avoid potential catastrophic failure, it is essential to detect damage to these composite structures at the earliest possible stage. Evolutionary approaches, such as genetic algorithms and simulated annealing, have attracted considerable attention as a means of solving a range of combinatorial optimization problems. Accordingly, the current study applies a hybrid evolutionary approach to detect damage in a simply supported equal-sided sector of a spherical laminate shell on the basis of natural frequency and mode shape data obtained from modal testing. The simulations consider two different damage scenarios, namely single-point damage and multiple-point damage. Furthermore, to simulate the measurement errors inherent in experimental modal testing, the simulations consider three specific error conditions, i.e. no error (the ideal case), a maximum 5% error in both the natural frequency and the mode shape data, and a maximum 5% error in the natural frequency data and a maximum 10% error in the mode shape data. The simulation results indicate that the algorithm successfully detects both the location and the extent of the damage in both damage scenarios irrespective of the magnitude of the errors assigned to the natural frequency and mode shape data.

Coalition formation mechanism in multi-agent systems based on genetic algorithms

As an important coordination and cooperation mechanism in multi-agent systems, coalition of agents exhibits some excellent characteristics and draws researchers’ attention increasingly. Cooperation formation has been a very active area of research in multi-agent systems. An efficient algorithm is needed for this topic since the numbers of the possible coalitions are exponential in the number of agents. Genetic algorithm (GA) has been widely reckoned as a useful tool for obtaining high quality and optimal solutions for a broad range of combinatorial optimization problems due to its intelligent advantages of self-organization, self-adaptation and inherent parallelism. This paper proposes a GA-based algorithm for coalition structure formation which aims at achieving goals of high performance, scalability, and fast convergence rate simultaneously. A novel 2D binary chromosome encoding approach and corresponding crossover and mutation operators are presented in this paper. Two valid parental chromosomes are certain to produce a valid offspring under the operation of the crossover operator. This improves the efficiency and shortens the running time greatly. The proposed algorithm is evaluated through a robust comparison with heuristic search algorithms. We have confirmed that our new algorithm is robust, self-adaptive and very efficient by experiments. The results of the proposed algorithm are found to be satisfactory.

Ant colony optimization for the examination scheduling problem

Ant colony optimization is an evolutionary search procedure based on the way that ant colonies cooperate in locating shortest routes to food sources. Early implementations focussed on the travelling salesman and other routing problems but it is now being applied to an increasingly diverse range of combinatorial optimization problems. This paper is concerned with its application to the examination scheduling problem. It builds on an existing implementation for the graph colouring problem to produce clash-free timetables and goes on to consider the introduction of a number of additional practical constraints and objectives. A number of enhancements and modifications to the original algorithm are introduced and evaluated. Results based on real-examination scheduling problems including standard benchmark data (the Carter data set) show that the final implementation is able to compete effectively with the best-known solution approaches to the problem.