Summary of Volume 3: Applications of Combinatorial Optimization

OF THE DISSERTATION Two applications of combinatorial optimization by Matthew R. Oster Dissertation Director: Professor Jonathan Eckstein This thesis presents two applications of combinatorial optimization. The first part contains a detailed description of a conference scheduling problem. We model the problem as a symmetric clustering problem, or a variant of minimum k-partition we call capacitated k-partition. This problem is proved to be NP-hard to solve to optimality, and further, unless P = NP, no constant factor polynomial-time approximation algorithm exists. We also propose a branch-and-cut algorithm with semidefinite programming relaxations enhanced with polyhedral cuts found at each tree node. Many cutting planes are demonstrated to be satisfied, or provably close to being satisfied, by semidefinite matrices in the variable space [−1/(k − 1), 1], which is in contrast to basic linear programming relaxations. Our algorithm also relies on a novel heuristic strategy when attempting to generate feasible solutions at every tree node. We test an implementation of our algorithm on random k-partition instances as well as a particular conference data set which comes from the 13th Annual INFORMS Computing Society Conference and was solved to within 0.85% of optimum in under 4 hours. The results here are promising and provide a starting point for future projects. In the second part, we describe a project called the Boat Allocation Module, where a team comprised of United States Coast Guard (USCG) analysts, and Command, Control, and Interoperability Center for Advanced Data Analysis researchers worked

As a well‐structured and costly activity that pervades industries in both the public and private sector, vehicle fleet management would appear to be a splendid candidate for model‐based planning and optimization. And yet, until recently the combinatorial intricacies of vehicle routing and of vehicle scheduling have precluded the widespread use of optimization (exact) methods for this problem class. Our discussion in this paper identifies the extent and nature of these problem complexities and draws contrasts with other applications of combinatorial optimization. It also summarizes a number of successful uses of optimization for vehicle fleet planning and highlights potentially fruitful avenues for algorithmic development. In particular, we describe several alternative models and novel algorithms for the vehicle routing problem, show how various modeling approaches for this problem are intimately related, and illustrate the interplay between model formulations and the algorithms that they suggest. This discussion shows that prospects for applying exact methods, possibly in conjunction with heuristics, are far from fully realized and points to vehicle fleet planning as a tempting target of opportunity for further investigation.

Finding the largest suborder of fixed width in a partially ordered set is an interesting combinatorial problem with applications in combinatorial optimization and scheduling. We present a polynomial time solution for this problem by transforming it into a minimum cost network flow problem in an appropriate auxiliary network.

This paper deals with algorithms for positive semidefinite matrix factorization (PSDMF). PSDMF is a recently-proposed extension of nonnegative matrix factorization with applications in combinatorial optimization, among others. In this paper, we focus on improving the local convergence of an alternating block gradient (ABC) method for PSDMF in a noise-free setting by replacing the quadratic objective function with the Poisson log-likelihood. This idea is based on truncated Wirtinger flow (TWF), a phase retrieval (PR) method that trims outliers in the gradient and thus regularizes it. Our motivation is a recent result linking PR with PSDMF. Our numerical experiments validate that the numerical benefits of TWF may carry over to PSDMF despite the more challenging setting, when initialized within its region of convergence. We then extend TWF from PR to affine rank minimization (ARM), and show that although the outliers are no longer an issue in the ARM setting, PSDMF with the new objective function may still achieves a smaller error for the same number of iterations. In a broader view, our results indicate that a proper choice of objective function may enhance convergence of matrix (or tensor) factorization methods.

PurposeThe purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.Design/methodology/approachIn the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.FindingsA new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.Originality/valueThe present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

The comparative performance of Integer Programming (IP) and Constraint Logic Programming (CLP) is explored by examining a number of models for four different combinatorial optimization applications. Computational results show contrasting behavior for the two approaches, and an analysis of performance with respect to problem and model characteristics is presented. The analysis shows that tightness of formulation is of great benefit to CLP where effective search reduction results in problems that can be solved quickly. In IP, if the linear feasible region does not identify the corresponding integer polytope, the problem may be difficult to solve. The paper identifies other characteristics of model behavior and concludes by examining ways in which IP and CLP may be incorporated within hybrid solvers.

Combinatorial optimization problems are becoming more and more important as its applications take place in many aspects, especially in logistics which planning and management of warehouse are mostly based on. Using exhaustive search for a precise solution is likely not preferable, since the complexity and execution time increases as factorial of input. So people attempt to look for a method which could provide a reasonably good solution with fast response, which heuristic and metaheuristic approaches are superior. In this paper, we will discuss briefly about some common heuristic and metaheuristic ones, as well as their principles and how they are applied to seek the optimal routes in a predefined graph. Then we come to a realistic application in logistics that AGVs have to allocate the best routes and order to achieve the tasks given, by simulation in MATLAB GUI program.

Submodular optimization generalizes many classic problems in combinatorial\noptimization and has recently found a wide range of applications in machine\nlearning (e.g., feature engineering and active learning). For many large-scale\noptimization problems, we are often concerned with the adaptivity complexity of\nan algorithm, which quantifies the number of sequential rounds where\npolynomially-many independent function evaluations can be executed in parallel.\nWhile low adaptivity is ideal, it is not sufficient for a distributed algorithm\nto be efficient, since in many practical applications of submodular\noptimization the number of function evaluations becomes prohibitively\nexpensive. Motivated by these applications, we study the adaptivity and query\ncomplexity of adaptive submodular optimization.\n Our main result is a distributed algorithm for maximizing a monotone\nsubmodular function with cardinality constraint $k$ that achieves a\n$(1-1/e-\\varepsilon)$-approximation in expectation. This algorithm runs in\n$O(\\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation\noracle in expectation. The approximation guarantee and query complexity are\noptimal, and the adaptivity is nearly optimal. Moreover, the number of queries\nis substantially less than in previous works. Last, we extend our results to\nthe submodular cover problem to demonstrate the generality of our algorithm and\ntechniques.\n

Eigenvalues and eigenvectors have many applications in structural mechanics and combinatorial optimization. In this paper, a set of matrices of special forms is studied for which the calculation of eigenvalues can be performed much easier than with the existing general methods. First tri-diagonal matrices are presented and then the relationships for calculating their eigenvalues are extended to the evaluation of the eigenvalues of block tri-diagonal matrices. Block penta-diagonal matrices are also studied in this paper. The eigensolution of different problems of structural mechanics is performed to show the simplicity of using the present formulations.

We reconsider the version of the traveling salesman problem (TSP) first studied in a well-known paper by Gilmore and Gomory (1964). In this, the distance between two cities A and B, is an integrable function of the x-coordinate of A and the y-coordinate of B. This problem finds important applications in machine scheduling, workforce planning, and combinatorial optimization. We solve this TSP variant by a {\mathcal O}(n log n) algorithm considerably simpler than previously known algorithms. The new algorithm demonstrates and exploits the structure of an optimal solution, and recreates it using minimal storage space without the use of edge interchanges.

Submodularity is one of the most well-studied properties of problem classes in combinatorial optimization and many applications of machine learning and data mining, with strong implications for guaranteed optimization. In this thesis, we investigate the role of submodularity in provable non-convex optimization and validation of algorithms. A profound understanding which classes of functions can be tractably optimized remains a central challenge for non-convex optimization. By advancing the notion of submodularity to domains (termed continuous submodularity), we characterize a class of generally non-convex and non-concave functions -- submodular functions, and derive algorithms for approximately maximizing them with strong approximation guarantees. Meanwhile, submodularity captures a wide spectrum of applications, ranging from revenue maximization with general marketing strategies, MAP inference for DPPs to mean field inference for probabilistic log-submodular models, which renders it as a valuable domain knowledge in optimizing this class of objectives. Validation of algorithms is an information-theoretic framework to investigate the robustness of algorithms to fluctuations in the input/observations and their generalization ability. We investigate various algorithms for one of the paradigmatic unconstrained submodular maximization problem: MaxCut. Due to submodularity of the MaxCut objective, we are able to present efficient approaches to calculate the algorithmic information content of MaxCut algorithms. The results provide insights into the robustness of different algorithmic techniques for MaxCut.

Combinatorial optimization problems have very important applications in information technology, transportation, economics, management, network communication, and other fields. Since the problem size in real-scenario application is in large-scale, the demand for real-time and efficient solving approaches increases rapidly. The traditional exact methods guarantee the optimality of the final solution, but these methods can hardly solve the problem in acceptable time due to extremely high computational costs. Heuristic approaches can find feasible solutions in a limited time, while these approaches cannot meet the demand of solution quality. In recent years, hybrid algorithms based on exact methods and heuristic algorithms show outstanding performance in solving large-scale combinatorial optimization problems. The hybridization not only overcomes the shortcomings from single algorithm but also fully utilizes the search ability for population-based approaches as well as the interpretability in exact methods, which promotes the application of combinatorial optimization in real-world problems. This paper reviews existing studies on hybrid algorithms combining exact method and evolutionary computation, summarizes the characteristics of the existing algorithms, and directs the future research.

Test case minimization approach using fault detection and combinatorial optimization techniques for configuration-aware structural testing

Heuristic search techniques are known for their efficiency and effectiveness in solving NP-hard problems. In this chapter, we present a heuristic search framework allowing the synthesis and evaluation of a variety of algorithms and also a library of oneway constraints for problem modelling. The integration of the two is explained in the context of the iOpt toolkit. iOpt is a large software system comprising several libraries and frameworks dedicated to the development of combinatorial optimization applications based on heuristic search.KeywordsLocal SearchTabu SearchHeuristic SearchGraph ColoringVehicle Rout ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The article is devoted to the issue of requirements fulfillment aimed at ensuring information security in an organization. The main contradiction of the subject area concerned is pointed out, which consists in the presence of a huge number of different variants of requirements fulfillment in the absence of a possibility to choose their correct and optimal order. The task of requirements ranking is set and the idea of the proposed solution is described in the form of seven provisions aimed at coordinated recording of heterogeneous requirements in a single notation, and an intuitive scheme of the idea is synthesized (with all seven provisions indicated on it).
 To represent the idea, the following entities are introduced: an object-organization and its elements to which requirements are imposed; generalized conditions for satisfying requirements that do not depend on the specifics of the organization; variations of sets of conditions that take into account a particular organization; basic conditions that check the presence/absence of elements 
 of the object and the values of their parameters; algorithms of activities in the organization to satisfy the conditions; priorities of requirements and resources needed by the algorithms. It is concluded that such formalization will lead organically to the algorithmic solution of the ranking problem and, eventually, to automation.
 The most suitable automated ways of solving the problem of ranking information security requirements – algorithmic application of combinatorial optimization and machine learning methods – are specified. Their high efficiency in comparison with «manual» methods used in modern information protection practice is predicted.
 The novelty, theoretical and practical significance of the obtained results are noted, as well as the prospect of further research – the construction of an analytical model of requirements fulfillment, which could be the basis of an appropriate method, followed by its program implementation and conducting of necessary experiments.