Combinatorial optimization (CO) is essential for improving efficiency and performance in engineering applications. Traditional algorithms based on pure mathematical reasoning are limited and incapable to capture the contextual nuances for optimization. This study explores the potential of Large Language Models (LLMs) in solving engineering CO problems by leveraging their reasoning power and contextual knowledge. We propose a novel LLM-based framework that integrates network topology and contextual domain knowledge to optimize the sequencing of Design Structure Matrix (DSM) —a common CO problem. Our experiments on various DSM cases demonstrate that the proposed method achieves faster convergence and higher solution quality than benchmark methods. Moreover, results show that incorporating contextual domain knowledge significantly improves performance despite the choice of LLMs.

Combinatorial optimization problems in the social and behavioral sciences are frequently associated with a variety of alternative objective criteria. Multiobjective programming is an operations research methodology that enables the quantitative analyst to investigate tradeoffs among relevant objective criteria. In this paper, we describe an interactive procedure for multiobjective asymmetric unidimensional seriation problems. This procedure uses a dynamic-programming algorithm to partially generate the efficient set of sequences for small to medium-sized problems, and a multioperation heuristic to estimate the efficient set for larger problems. The interactive multiobjective procedure is applied to an empirical data set from the psychometric literature. We conclude with a discussion of other potential areas of application in combinatorial data analysis.

In educational practice, a test assembly problem is formulated as a system of inequalities induced by test specifications. Each solution to the system is a test, represented by a 0–1 vector, where each element corresponds to an item included (1) or not included (0) into the test. Therefore, the size of a 0–1 vector equals the number of items n in a given item pool. All solutions form a feasible set—a subset of 2n vertices of the unit cube in an n-dimensional vector space. Test assembly is uniform if each test from the feasible set has an equal probability of being assembled. This paper demonstrates several important applications of uniform test assembly for educational practice. Based on Slepian’s inequality, a binary program was analytically studied as a candidate for uniform test assembly. The results of this study establish a connection between combinatorial optimization and probability inequalities. They identify combinatorial properties of the feasible set that control the uniformity of the binary programming test assembly. Computer experiments illustrating the concepts of this paper are presented.

In many regression applications, users are often faced with difficulties due to nonlinear relationships, heterogeneous subjects, or time series which are best represented by splines. In such applications, two or more regression functions are often necessary to best summarize the underlying structure of the data. Unfortunately, in most cases, it is not known a priori which subset of observations should be approximated with which specific regression function. This paper presents a methodology which simultaneously clusters observations into a preset number of groups and estimates the corresponding regression functions' coefficients, all to optimize a common objective function. We describe the problem and discuss related procedures. A new simulated annealing-based methodology is described as well as program options to accommodate overlapping or nonoverlapping clustering, replications per subject, univariate or multivariate dependent variables, and constraints imposed on cluster membership. Extensive Monte Carlo analyses are reported which investigate the overall performance of the methodology. A consumer psychology application is provided concerning a conjoint analysis investigation of consumer satisfaction determinants. Finally, other applications and extensions of the methodology are discussed.

We present a new model and associated algorithm, INDCLUS, that generalizes the Shepard-Arabie ADCLUS (ADditive CLUStering) model and the MAPCLUS algorithm, so as to represent in a clustering solution individual differences among subjects or other sources of data. Like MAPCLUS, the INDCLUS generalization utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach to impose discrete (0,1) constraints on parameters defining cluster membership. All subjects in an INDCLUS analysis are assumed to have a common set of clusters, which are differentially weighted by subjects in order to portray individual differences. As such, INDCLUS provides a (discrete) clustering counterpart to the Carroll-Chang INDSCAL model for (continuous) spatial representations. Finally, we consider possible generalizations of the INDCLUS model and algorithm.

The selection of a subset of variables from a pool of candidates is an important problem in several areas of multivariate statistics. Within the context of principal component analysis (PCA), a number of authors have argued that subset selection is crucial for identifying those variables that are required for correct interpretation of the components. In this paper, we adapt the variable neighborhood search (VNS) paradigm to develop two heuristics for variable selection in PCA. The performances of these heuristics were compared to those obtained by a branch-and-bound algorithm, as well as forward stepwise, backward stepwise, and tabu search heuristics. In the first experiment, which considered candidate pools of 18 to 30 variables, the VNS heuristics matched the optimal subset obtained by the branch-and-bound algorithm more frequently than their competitors. In the second experiment, which considered candidate pools of 54 to 90 variables, the VNS heuristics provided better solutions than their competitors for a large percentage of the test problems. An application to a real-world data set is provided to demonstrate the importance of variable selection in the context of PCA.

The seriation of proximity matrices is an important problem in combinatorial data analysis and can be conducted using a variety of objective criteria. Some of the most popular criteria for evaluating an ordering of objects are based on (anti-) Robinson forms, which reflect the pattern of elements within each row and/or column of the reordered matrix when moving away from the main diagonal. This paper presents a branch-and-bound algorithm that can be used to seriate a symmetric dissimilarity matrix by identifying a reordering of rows and columns of the matrix optimizing an anti-Robinson criterion. Computational results are provided for several proximity matrices from the literature using four different anti-Robinson criteria. The results suggest that with respect to computational efficiency, the branch-and-bound algorithm is generally competitive with dynamic programming. Further, because it requires much less storage than dynamic programming, the branch-and-bound algorithm can provide guaranteed optimal solutions for matrices that are too large for dynamic programming implementations.

A recursive dynamic programming strategy is discussed for optimally reorganizing the rows and simultaneously the columns of an n × n proximity matrix when the objective function measuring the adequacy of a reorganization has a fairly simple additive structure. A number of possible objective functions are mentioned along with several numerical examples using Thurstone's paired comparison data on the relative seriousness of crime. Finally, the optimization tasks we propose to attack with dynamic programming are placed in a broader theoretical context of what is typically referred to as the quadratic assignment problem and its extension to cubic assignment.

This paper develops a stochastic model for comparing payments to U.S. corn producers under the U.S. Senate's Average Crop Revenue Program (ACR) versus payments under the price-based marketing loan benefit and countercyclical payment programs. Using this model, the paper examines the sensitivity of the density function for payments to changes in expected price levels. We also assess the impact of the choice of yield aggregation used in the ACR payment rate on the mean and variance of farm returns. We find that ACR payments lower the producer's coefficient of variation of total revenue more than does the price-based support, although ACR may not raise mean revenue as much. While corn farmers in the heartland states might still prefer to receive the traditional forms of support when prices are low relative to statutory loan rates and target prices, this outcome is not necessarily the case for farmers in peripheral production regions.

Image fusion is an imaging technique to visualize information from multiple imaging sources in one single image, which is widely used in remote sensing, medical imaging etc. In this work, we study two variational approaches to image fusion which are closely related to the standard TV-L 2 and TV-L 1 image approximation methods. We investigate their convex optimization formulations, under the perspective of primal and dual, and propose their associated new image decomposition models. In addition, we consider the TV-L 1 based image fusion approach and study the specified problem of fusing two discrete-constrained images and where and are the sets of linearly-ordered discrete values. We prove that the TV-L 1 based image fusion actually gives rise to the exact convex relaxation to the corresponding nonconvex image fusion constrained by the discrete-valued set This extends the results for the global optimization of the discrete-constrained TV-L 1 image approximation [8, 36] to the case of image fusion. As a big numerical advantage of the two proposed dual models, we show both of them directly lead to new fast and reliable algorithms, based on modern convex optimization techniques. Experiments with medical images, remote sensing images and multi-focus images visibly show the qualitative differences between the two studied variational models of image fusion. We also apply the new variational approaches to fusing 3D medical images.

A novel swarm intelligence approach for combinatorial optimization is proposed, which we call probability increment based swarm optimization (PIBSO). The population evolution mechanism of PIBSO is depicted. Each state in search space has a probability to be chosen. The rule of increasing the probabilities of states is established. Incremental factor is proposed to update probability of a state, and its value is determined by the fitness of the state. It lets the states with better fitness have higher probabilities. Usual roulette wheel selection is employed to select states. Population evolution is impelled by roulette wheel selection and state probability updating. The most distinctive feature of PIBSO is because roulette wheel selection and probability updating produce a trade-off between global and local search; when PIBSO is applied to solve the printed circuit board assembly optimization problem (PCBAOP), it performs superiorly over existing genetic algorithm and adaptive particle swarm optimization on length of tour and CPU running time, respectively. The reason for having such advantages is analyzed in detail. The success of PCBAOP application verifies the effectiveness and efficiency of PIBSO and shows that it is a good method for combinatorial optimization in engineering.

The traveling salesman problem (TSP) is one of the most fundamental optimizationproblems. We consider the β-metric traveling salesman problem(Δβ-TSP), i.e., the TSPrestricted to graphs satisfying the β-triangle inequalityc({v,w}) ≤ β(c({v,u}) + c({u,w})),for some cost function c and any three vertices u,v,w.The well-known path matching Christofides algorithm (PMCA) guarantees an approximationratio of 3β2/2 and is the best known algorithm for theΔβ-TSP, for 1 ≤ β ≤ 2. Weprovide a complete analysis of the algorithm. First, we correct an error in the originalimplementation that may produce an invalid solution. Using a worst-case example, we thenshow that the algorithm cannot guarantee a better approximation ratio. The example canalso be used for the PMCA variants for the Hamiltonian path problem with zero and oneprespecified endpoints. For two prespecified endpoints, we cannot reuse the example, butwe construct another worst-case example to show the optimality of the analysis also inthis case.

In this paper we present a new approach to solve the Minimum Independent Dominating Setproblem in general graphs which is one of the hardest optimization problem. We propose amethod using a clique partition of the graph, partition that can be obtained greedily. Weprovide conditions under which our method has a better complexity than the complexity ofthe previously known algorithms. Based on our theoretical method, we design in the secondpart of this paper an efficient algorithm by including cuts in the search process. We thenexperiment it and show that it is able to solve almost all instances up to 50 vertices inreasonable time and some instances up to several hundreds of vertices. To go further andto treat larger graphs, we analyze a greedy heuristic. We show that it often gives good(sometimes optimal) results in large instances up to 60   000 vertices in less than 20 s.That sort of heuristic is a good approach to get an initial solution for our exact method.We also describe and analyze some of its worst cases.

We consider the scheduling of an interval order precedence graph of unit execution time tasks with communication delays, release dates and deadlines. Tasks must be executed by a set of processors partitioned into K classes; each task requires one processor from a fixed class. The aim of this paper is to study the extension of the Leung–Palem–Pnueli (in short LPP) algorithm to this problem. The main result is to prove that the LPP algorithm can be extended to dedicated processors and monotone communication delays. It is also proved that the problem is NP–complete for two dedicated processors if communication delays are non monotone. Lastly, we show that list scheduling algorithm cannot provide a solution for identical processors.

Uncertainty in optimization is not a new ingredient. Diverse modelsconsidering uncertainty have been developed over the last 40 years.In our paper we essentially discuss a particular uncertainty modelassociated with combinatorial optimization problems, developed inthe 90's and broadly studied in the past years. This approach namedminmax regret (in particular our emphasis is on the robustdeviation criteria) is different from the classical approach for handlinguncertainty, stochastic approach, where uncertainty is modeledby assumed probability distributions over the space of all possiblescenarios and the objective is to find a solution with good probabilisticperformance. In the minmax regret (MMR) approach, the set of all possible scenariosis described deterministically, and the search is for a solution thatperforms reasonably well for all scenarios, i.e., that has the bestworst-case performance. In this paper we discuss the computational complexity of some classiccombinatorial optimization problems using MMR approach, analyze thedesign of several algorithms for these problems, suggest the studyof some specific research problems in this attractive area, and alsodiscuss some applications using this model.

Quantum annealing, or quantum stochastic optimization, is a classical randomized algorithm which provides good heuristics for the solution of hard optimization problems. The algorithm, suggested by the behaviour of quantum systems, is an example of proficuous cross contamination between classical and quantum computer science. In this survey paper we illustrate how hard combinatorial problems are tackled by quantum computation and present some examples of the heuristics provided by quantum annealing. We also present preliminary results about the application of quantum dissipation (as an alternative to imaginary time evolution) to the task of driving a quantum system toward its state of lowest energy.

This paper addresses a combinatorial optimization problem (COP), namely a variant of the (standard) matrix chain product (MCP) problem where the matrices are square and either dense (i.e. full) or lower/upper triangular. Given a matrix chain of length n, we first present a dynamic programming algorithm (DPA) adapted from the well known standard algorithm and having the same O(n 3) complexity. We then design and analyse two optimal O(n) greedy algorithms leading in general to different optimal solutions i.e. chain parenthesizations. Afterwards, we establish a comparison between these two algorithms based on the parallel computing of the matrix chain product through intra and inter-subchains coarse grain parallelism. Finally, an experimental study illustrates the theoretical parallel performances of the designed algorithms.

Many combinatorial optimization problems can be formulated asthe minimization of a 0–1 quadratic function subject to linear constraints. Inthis paper, we are interested in the exact solution of this problem through atwo-phase general scheme. The first phase consists in reformulating theinitial problem either into a compact mixed integer linear program or into a0–1 quadratic convex program. The second phase simply consists insubmitting the reformulated problem to a standard solver. The efficiency ofthis scheme strongly depends on the quality of the reformulation obtained inphase 1. We show that a good compact linear reformulation can be obtained bysolving a continuous linear relaxation of the initial problem. We also showthat a good quadratic convex reformulation can be obtained by solving asemidefinite relaxation. In both cases, the obtained reformulation profitsfrom the quality of the underlying relaxation. Hence, the proposed scheme getsaround, in a sense, the difficulty to incorporate these costly relaxations ina branch-and-bound algorithm.

The NP-hard problem of car sequencing has received a lot of attention these last years. Whereas a directapproach based on integer programming or constraint programming is generally fruitless when the number of vehicles tosequence exceeds the hundred, several heuristics have shown their efficiency. In this paper, very large-scaleneighborhood improvement techniques based on integer programming and linear assignment are presented for solving carsequencing problems. The effectiveness of this approach is demonstrated through an experimental study made on seminalCSPlib's benchmarks.

In the paper, the problem of the genome mapping of DNA molecules, is presented. In particular, the new approach — the Simplified Partial Digest Problem (SPDP), is analyzed. This approach, although easy in laboratory implementation and robust with respect to measurement errors, when formulated in terms of a combinatorial search problem, is proved to be strongly NP-hard for the general error-free case. For a subproblem of the SPDP, a simple O(nlogn)-time algorithm is given, where n is a number of restriction sites.