Maximum Diversity Problem Research Articles

A method for the systematic selection of enzyme panel candidates by solving the maximum diversity problem

Enzymes are being increasingly exploited for their potential as industrial biocatalysts. Establishing a portfolio of useful biocatalysts from large and diverse protein family is challenging and a systematic method for candidate selection promises to aid in this task. Moreover, accurate enzyme functional annotation can only be confidently guaranteed through experimental characterisation in the laboratory. The selection of catalytically diverse enzyme panels for experimental characterisation is also an important step for shedding light on the currently unannotated proteins in enzyme families. Current selection methods often lack efficiency and scalability, and are usually non-systematic. We present a novel algorithm for the automatic selection of subsets from enzyme families. A tabu search algorithm solving the maximum diversity problem for sequence identity was designed and implemented, and applied to three diverse enzyme families. We show that this approach automatically selects panels of enzymes that contain high richness and relative abundance of the known catalytic functions, and outperforms other methods such as k-medoids.

New advances for quantum‐inspired optimization

AbstractAdvances in quantum computing with applications in combinatorial optimization have evolved at an increasing rate in recent years. The quadratic unconstrained binary optimization (QUBO) model is at the center of these developments and has become recognized as an effective alternative method for representing a wide variety of combinatorial optimization problems. Additional momentum has resulted from the arrival of quantum computers and their ability to solve the Ising spin glass problem, another form of the QUBO model. This paper highlights advances in solving QUBO models and extensions to more general polynomial unconstrained binary optimization (PUBO) models as important alternatives to traditional approaches. Computational experience is provided that compares the performance of unique quantum‐inspired metaheuristic solvers—the Next Generation Quantum (NGQ) solver for QUBO models and the NGQ‐PUBO solver for PUBO models—with the performance of CPLEX and the Dwave quantum advantage solver. Extensive results, including experiments with a set of large set partitioning problems representing the largest QUBO models reported in the literature to date, along with maximum diversity and max cut problem sets, disclose that our solvers outperform both CPLEX and Dwave by a wide margin in terms of both computational time and solution quality.

A Learnheuristic Algorithm for the Capacitated Dispersion Problem under Dynamic Conditions

The capacitated dispersion problem, which is a variant of the maximum diversity problem, aims to determine a set of elements within a network. These elements could symbolize, for instance, facilities in a supply chain or transmission nodes in a telecommunication network. While each element typically has a bounded service capacity, in this research, we introduce a twist. The capacity of each node might be influenced by a random Bernoulli component, thereby rendering the possibility of a node having zero capacity, which is contingent upon a black box mechanism that accounts for environmental variables. Recognizing the inherent complexity and the NP-hard nature of the capacitated dispersion problem, heuristic algorithms have become indispensable for handling larger instances. In this paper, we introduce a novel approach by hybridizing a heuristic algorithm with reinforcement learning to address this intricate problem variant.

Biased Random-Key Genetic Algorithm with Local Search Applied to the Maximum Diversity Problem

The maximum diversity problem (MDP) aims to select a subset with a predetermined number of elements from a given set, maximizing the diversity among them. This NP-hard problem requires efficient algorithms that can generate high-quality solutions within reasonable computational time. In this study, we propose a novel approach that combines the biased random-key genetic algorithm (BRKGA) with local search to tackle the MDP. Our computational study utilizes a comprehensive set of MDPLib instances, and demonstrates the superior average performance of our proposed algorithm compared to existing literature results. The MDP has a wide range of practical applications, including biology, ecology, and management. We provide future research directions for improving the algorithm’s performance and exploring its applicability in real-world scenarios.

A Multi-Start Biased-Randomized Algorithm for the Capacitated Dispersion Problem

The capacitated dispersion problem is a variant of the maximum diversity problem in which a set of elements in a network must be determined. These elements might represent, for instance, facilities in a logistics network or transmission devices in a telecommunication network. Usually, it is considered that each element is limited in its servicing capacity. Hence, given a set of possible locations, the capacitated dispersion problem consists of selecting a subset that maximizes the minimum distance between any pair of elements while reaching an aggregated servicing capacity. Since this servicing capacity is a highly usual constraint in real-world problems, the capacitated dispersion problem is often a more realistic approach than is the traditional maximum diversity problem. Given that the capacitated dispersion problem is an NP-hard problem, whenever large-sized instances are considered, we need to use heuristic-based algorithms to obtain high-quality solutions in reasonable computational times. Accordingly, this work proposes a multi-start biased-randomized algorithm to efficiently solve the capacitated dispersion problem. A series of computational experiments is conducted employing small-, medium-, and large-sized instances. Our results are compared with the best-known solutions reported in the literature, some of which have been proven to be optimal. Our proposed approach is proven to be highly competitive, as it achieves either optimal or near-optimal solutions and outperforms the non-optimal best-known solutions in many cases. Finally, a sensitive analysis considering different levels of the minimum aggregate capacity is performed as well to complete our study.

A maximum diversity-based path sparsification for geometric graph matching

This paper presents an effective dissimilarity measure for geometric graphs representing shapes. The proposed dissimilarity measure is a distance that combines a sparsification of the geometric graph based on the maximum diversity problem and a new node embedding that captures the topological neighborhood of nodes. The sparsification step aims to reduce the size of the graph and to correct the misdistribution of nodes on the geometric graph induced by the noise of image handling. Experimental evaluation shows that the sparsification algorithm retains the form of the shapes while decreasing the number of processed nodes which reduces the overall matching time. Furthermore, the proposed node embedding and similarity measure give better performance in comparison with existing graph matching approaches.

Multi-Objective GRASP for Maximizing Diversity

This work presents a novel greedy randomized adaptive search procedure approach for dealing with the maximum diversity problem from a multi-objective perspective. In particular, five of the most extended diversity metrics were considered, with the aim of maximizing all of them simultaneously. The metrics considered have been proven to be in conflict, i.e., it is not possible to optimize one metric without deteriorating another one. Therefore, this results in a multi-objective optimization problem where a set of efficient solutions that are diverse with respect to all the metrics at the same time must be obtained. A novel adaptation of the well-known greedy randomized adaptive search procedure, which has been traditionally used for single-objective optimization, was proposed. Two new constructive procedures are presented to generate a set of efficient solutions. Then, the improvement phase of the proposed algorithm consists of a new efficient local search procedure based on an exchange neighborhood structure that follows a first improvement approach. An effective exploration of the exchange neighborhood structure is also presented, to firstly explore the most promising ones. This feature allowed the local search proposed to limit the size of the neighborhood explored, resulting in an efficient exploration of the solution space. The computational experiments showed the merit of the proposed algorithm, when comparing the obtained results with the best previous method in the literature. Additionally, new multi-objective evolutionary algorithms derived from the state-of-the-art were also included in the comparison, to prove the quality of the proposal. Furthermore, the differences found were supported by non-parametric statistical tests.

T-Linearization for the maximum diversity problem

We apply the t-linearization technique to the maximum diversity problem (MDP) and compare its performance with other well-known linearizations. We lift the t-constraints based on the cardinality restriction, derive valid inequalities for MDP, and show their usefulness to solve the problem within the t-linearization framework. We propose and computationally evaluate a branch-and-bound algorithm on benchmark instances. The algorithm is compared with other exact approaches from the literature.

A two-phase tabu search based evolutionary algorithm for the maximum diversity problem

In this paper, we study the maximum diversity problem (MDP) which is equivalent to the quadratic unconstrained binary optimization (QUBO) problem with cardinality constraint. The MDP aims to select a subset of elements with given cardinality such that the sum of pairwise distances between any two elements in the selected subset is maximized. For solving this computationally challenging problem, we propose a two-phase tabu search based evolutionary algorithm (TPTS/EA), which integrates several distinguishing features to ensure the diversity and the quality of the evolution, such as a two-phase tabu search algorithm which consists of a dynamic candidate list (DCL) strategy-based traditional tabu search in the first phase and a solution-based tabu search procedure to refine the search in the second phase, and two path-relinking based recombination operators to generate new offspring solutions. Tested on three sets of totally 140 public instances in the literature, the study demonstrates the efficacy of the proposed TPTS/EA algorithm in terms of both solution quality and computational efficiency. Specifically, our proposed TPTS/EA algorithm is able to improve the previous best known results for 2 instances, while matching the previous best-known solutions for 130 instances. We also provide experimental evidences to highlight the beneficial effect of several important components in our TPTS/EA algorithm.

Maximum Diversity Problem. A Multi-Objective Approach

The Maximum Diversity (MD) problem is the process of selecting a subset of elements where the diversity among selected elements is maximized. Several diversity measures were already studied in the literature, optimizing the problem considered in a pure mono-objective approach. This work presents for the first time multi-objective approaches for the MD problem, considering the simultaneous optimization of the following five diversity measures: (i) Max-Sum, (ii) Max-Min, (iii) Max-MinSum, (iv) Min-Diff and (v) Min-P-center. Two different optimization models are proposed: (i) Multi-Objective Maximum Diversity (MMD) model, where the number of elements to be selected is defined a-priori, and (ii) Multi-Objective Maximum Average Diversity (MMAD) model, where the number of elements to be selected is also a decision variable. To solve the formulated problems, a Multi-Objective Evolutionary Algorithm (MOEA) is presented. Experimental results demonstrate that the proposed MOEA found good quality solutions, i.e. between 98.85% and 100% of the optimal Pareto front when considering the hypervolume for comparison purposes.

Opposition-Based Memetic Search for the Maximum Diversity Problem

As a usual model for a variety of practical applications, the maximum diversity problem (MDP) is computational challenging. In this paper, we present an opposition-based memetic algorithm (OBMA) for solving MDP, which integrates the concept of opposition-based learning (OBL) into the well-known memetic search framework. OBMA explores both candidate solutions and their opposite solutions during its initialization and evolution processes. Combined with a powerful local optimization procedure and a rank-based quality-and-distance pool updating strategy, OBMA establishes a suitable balance between exploration and exploitation of its search process. Computational results on 80 popular MDP benchmark instances show that the proposed algorithm matches the best-known solutions for most of instances, and finds improved best solutions (new lower bounds) for 22 instances. We provide experimental evidences to highlight the beneficial effect of OBL for solving MDP.

A note on heuristic approach based on UBQP formulation of the maximum diversity problem

The maximum diversity problem (MDP) is a challenging NP-hard problem with a wide range of real applications. Several researchers have pointed out close relationship between the MDP and unconstrained binary quadratic program (UBQP). In this paper, we provide procedures to solve MDP ideas from the UBQP formulation of the problem. We first give some local optimality results for r-flip improvement procedures on MDP. Then, a set of highly effective diversification approaches based on sequential improvement steps for MDP are presented. Four versions of the approaches are used within a simple tabu search and applied to 140 benchmark MDP problems available on the Internet. The procedures solve all 80 small- to medium-sized problems instantly to the best known solutions. For 22 of the 60 large problems, the procedures improved by significant amounts the best known solutions in reasonably short CPU time.

Solving the maximum min-sum dispersion by alternating formulations of two different problems

The maximum min-sum dispersion problem aims to maximize the minimum accumulative dispersion among the chosen elements. It is known to be strongly NP-hard problem. In this paper we present heuristic where the objective functions of two different problems are shifted within variable neighborhood search framework. Though this heuristic can be seen as an extended variant of variable formulation search approach that takes into account alternative formulations of one problem, the important difference is that it allows using alternative formulations of more than one optimization problem. Here we use one alternative formulation that is of max-sum type of the originally max–min type maximum diversity problem. Computational experiments on the benchmark instances used in the literature show that the suggested approach improves the best known results for most instances in a shorter computing time.

Solving wind farm layout optimization with mixed integer programs and constraint programs

The wind farm layout optimization problem is concerned with the optimal location of turbines within a fixed geographical area to maximize profit under stochastic wind conditions. Previously, it has been modeled as a maximum diversity (or $$p$$ -dispersion-sum) problem, but such a formulation cannot capture the nonlinearity of aerodynamic interactions among multiple wind turbines. We present the first constraint programming (CP) and mixed integer linear programming (MIP) models that incorporate such nonlinearity. Our empirical results indicate that the relative performance between these two models reverses when the wind scenario changes from a simple to a more complex one. We then extend these models to include landowner participation and noise constraints. With the additional constraints, the MIP-based decomposition outperforms CP in almost all cases. We also propose an improvement to the previous maximum diversity model and demonstrate that the improved model solves more problem instances.

Variable neighborhood search for maximum diverse grouping problem

This paper presents a general variable neighborhood search (GVNS) heuristic for solving the maximum diverse grouping problem. Extensive computational experiments performed on a series of large random graphs as well as on several instances of the maximum diversity problem taken from the literature show that the results obtained by GVNS consistently outperform the best heuristics from the literature.

A tabu search based memetic algorithm for the maximum diversity problem

This paper presents a highly effective memetic algorithm for the maximum diversity problem based on tabu search. The tabu search component uses a successive filter candidate list strategy and the solution combination component employs a combination operator based on identifying strongly determined and consistent variables. Computational experiments on three sets of 40 popular benchmark instances indicate that our tabu search/memetic algorithm (TS/MA) can easily obtain the best known results for all the tested instances (where no previous algorithm has achieved) as well as improved results for six instances. Analysis of comparisons with state-of-the-art algorithms demonstrates statistically that our TS/MA competes very favorably with the best performing algorithms. Key elements and properties of TS/MA are also analyzed to disclose the benefits of integrating tabu search (using a successive filter candidate list strategy) and solution combination (based on critical variables).

A hybrid metaheuristic method for the Maximum Diversity Problem

The Maximum Diversity Problem (MDP) consists in selecting a subset of m elements from a given set of n elements (n>m) in such a way that the sum of the pairwise distances between the m chosen elements is maximized. We present a hybrid metaheuristic algorithm (denoted by MAMDP) for MDP. The algorithm uses a dedicated crossover operator to generate new solutions and a constrained neighborhood tabu search procedure for local optimization. MAMDP applies also a distance-and-quality based replacement strategy to maintain population diversity. Extensive evaluations on a large set of 120 benchmark instances show that the proposed approach competes very favorably with the current state-of-art methods for MDP. In particular, it consistently and easily attains all the best known lower bounds and yields improved lower bounds for 6 large MDP instances. The key components of MAMDP are analyzed to shed light on their influence on the performance of the algorithm.

Memetic self-adaptive evolution strategies applied to the maximum diversity problem

The maximum diversity problem consists in finding a subset of elements which have maximum diversity between each other. It is a very important problem due to its general aspect, that implies many practical applications such as facility location, genetics, and product design. We propose a method based on evolution strategies with local search and self-adaptation of the parameters. For all time limits from 1 to 300 s as well as for time to converge to the best solutions known, this method leads to better results when compared to other state-of-the-art algorithms.

A hybrid data mining GRASP with path-relinking

The exploration of hybrid metaheuristics—combination of metaheuristics with concepts and processes from other research areas—has been an important trend in combinatorial optimization research. An instance of this study is the hybrid version of the GRASP metaheuristic that incorporates a data mining process. Traditional GRASP is an iterative metaheuristic which returns the best solution reached over all iterations. In the hybrid GRASP proposal, after executing a significant number of iterations, the data mining process extracts patterns from an elite set of sub-optimal solutions for the optimization problem. These patterns present characteristics of near optimal solutions and can be used to guide the following GRASP iterations in the search through the combinatorial solution space. The hybrid data mining GRASP has been successfully applied for different combinatorial problems: the set packing problem, the maximum diversity problem, the server replication for reliable multicast problem and the p-median problem. In this work, we show that, not only the traditional GRASP, but also GRASP improved with the path-relinking heuristic—a memory-based intensification strategy—could benefit from exploring a data mining procedure. Computational experiments, comparing traditional GRASP with path-relinking and different path-relinking hybrid proposals, showed that employing the combination of path-relinking and data mining made the GRASP find better results in less computational time. Another contribution of this work is the application of the path-relinking hybrid proposal for the 2-path network design problem, which improved the state-of-the-art solutions for this problem.

Adaptive GRASP for the maximum diversity problem

The problem of choosing a subset of elements with maximum diversity from a given set is known as the maximum diversity problem. GRASP is a multi-start method that consists of two phases: a solution construction phase, which randomly constructs a greedy solution, and an improvement phase, which uses that solution as an initial starting point. In the last few years, the GRASP methodology has arisen as a prospective metaheuristic approach to find high-quality solutions for this difficult problem in reasonable computational times. With the aim of providing additional results and insights along this line of research, this paper proposes a new GRASP model with two innovative components: a customized iterated greedy metaheuristic that acts as an improvement method and an adaptive construction phase that creates new solutions, incorporating pieces of the best solution found so far (the number of elements forming these pieces increases as time goes by), in this way balancing the diversification and intensification kept by GRASP. The benefits of the proposal in comparison to other metaheuristic proposed in the literature to deal with the problem are experimentally shown.