Properties of Some Combinatorial Optimization Problems and Their Effect on the Performance of Integer Programming and Constraint Logic Programming

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

Constraint Logic Programming (CLP) languages extend logic programming by allowing the use of constraints from different domains such as real numbers or Boolean functions. They have proved to be ideal for expressing problems that require interactive mathematical modeling and complex combinatorial optimization problems. However, CLP languages have mainly been considered as research systems, useful for rapid prototyping, by not really competitive with more conventional programming languages where efficiency is a more important consideration. One promising approach to improving the performance of CLP systems is the use of powerful program optimizations to reduce the cost of constraint solving. We extend work in this area by describing a new optimizing compiler for the CLP language CLP(ℛ). The compiler implements six powerful optimizations: reordering of constraints, removal of redundant variables, and specialization of constraints which cannot fail. Each program optimization is designed to remove the overhead of constraint solving when possible and keep the number of constraints in the store as small as possible. We systematically evaluate the effectiveness of each optimization in isolation and in combination. Our empirical evaluation of the compiler verifies that optimizing compilation can be made efficient enough to allow compilation of real-world programs and that it is worth performing such compilation because it gives significant time and space performance improvements.

Contributors

We propose a unifying method for analysis of scheduling problems in real-time systems. The method is based on ACSR-VP, a real-time process algebra with value-passing capabilities. We use ACSR-VP to describe an instance of a scheduling problem as a process that has parameters of the problem as free variables. The specification is analyzed by means of a symbolic algorithm. The outcome of the analysis is a set of equations, a solution to which yields the values of the parameters that make the system schedulable. Equations are solved using integer programming or constraint logic programming. The paper presents specifications of two scheduling problems as examples.

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.

Contributors

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.

Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.KeywordsInterior PointInterior Point MethodLinear Programming RelaxationInteger Programming ProblemInterior Point AlgorithmThese 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.

Most universities in Belgium do research in AI and offer AI courses to students. The largest part of the research is into the more classical AI subjects, such as logic programming and theorem proving. However, sub-symbolic AI is also investigated. It seems that all Belgian Universities have their own distinct flavour of research. Below, we present the different Belgian universities in alphabetical order. The Facultes Universitaires Notre Dame de la Paix Namur (University of Namur) has a group called ‘semantics and computational logic’ led by B. Le Charlier and P. Schobbens. This group is theoretically oriented and although their interests are broader than just AI, some of their research is into computational logic, automated theorem proving and formalisation of agentbased systems. The Katholieke Universiteit Leuven (Flemish catholic university of Leuven) has different groups working on artificial-intelligence subjects. First of all, there is the group on declarative languages and artificial intelligence led by M. Bruynooghe. The artificialintelligence interest of this group is into knowledge representation and machine learning, with research themes such as data mining, constraint logic programming, reasoning with time, natural-language processing and learning in agents. They are also involved in the Robocup simulation league. Second, there is a group doing research into intelligent mobile robots led by H. Van Brussel and J. De Schutter. This group is part of the mechanical engineering department. Finally, there is a group at the economy department, led by J. Vanthienen and M. Verhelst, that investigates knowledge systems and modelling. At the Limburgs Universitair Centrum (Limburg University Centre) M. Gyssens of the theoretical computer science group works on the structure and complexity of constraint satisfaction problems. J. Van den Bussche of the same group works on implementation of query languages for data mining. The Universite Catholique de Louvain (French catholic university of Louvain) has several small groups investigating different AI subjects. Computational linguistics is investigated by E. Milgrom and S. Granger. Logic programming is investigated by Y. Deville. A. Thayse investigates natural-language processing, P.J. Courtois investigates self-organising systems, P. Van Hentenryck works on optimisation and constraint satisfaction and Y. Kamp investigates collective intelligence. The Universite de Liege (Montefiore) (University of Liege) has D. Gribomont working on verification methods, automatic theorem proving and formal logic. The Universitaire Instelling Antwerpen (University of Antwerp) has the centre for Dutch language and speech, led by W. Daelemans. Part of the research of this institute is purely linguistic, but a great deal of research is also performed into (memory-based) machine learning of natural language, using large corpora of text, as well as into intelligent processing of text. The Universit Libre de Bruxelles (French Free University of Brussels) has the IRIDIA (Institut de Recherches Interdisciplinaires et de Developpements en Intelligence Artificielle) group led by M. Dorigo and H. Bersini. The main research interests are: ‘belief representation and AI-techniques for process control and classification’ and ‘nature-inspired heuristics for the solution of combinatorial and continuous space optimisation problems’. Sub-symbolic techniques are

Minimum cut problems are among the most classical problems in Combinatorial Optimization and are used in a wide set of applications. Some of the best-known efficiently solvable variants include global minimum cuts, minimum s-t cuts, and minimum odd cuts in undirected graphs. We study a problem class that can be seen to generalize the above variants, namely finding congruency-constrained minimum cuts, i.e., we consider cuts whose number of vertices is congruent to r modulo m, for some integers r and m. Apart from being a natural generalization of odd cuts, congruency-constrained minimum cuts exhibit an interesting link to a long-standing open problem in Integer Programming, namely whether integer programs described by an integer constraint matrix with bounded subdeterminants can be solved efficiently.

Decision tree induction techniques attempt to find small trees that fit a training set of data. This preference for smaller trees, which provides a learning bias, is often justified as being consistent with the principle of Occam’s Razor. Informally, this principle states that one should prefer the simpler hypothesis. In this paper we take this principle to the extreme. Specifically, we formulate decision tree induction as a combinatorial optimisation problem in which the objective is to minimise the number of nodes in the tree. We study alternative formulations based on satisfiability, constraint programming, and hybrids with integer linear programming. We empirically compare our approaches against standard induction algorithms, showing that the decision trees we obtain can sometimes be less than half the size of those found by other greedy methods. Furthermore, our decision trees are competitive in terms of accuracy on a variety of well-known benchmarks, often being the most accurate. Even when post-pruning of greedy trees is used, our constraint-based approach is never dominated by any of the existing techniques.

FOUNDATIONS. The Scope of Integer and Combinatorial Optimization. Linear Programming. Graphs and Networks. Polyhedral Theory. Computational Complexity. Polynomial-Time Algorithms for Linear Programming. Integer Lattices. GENERAL INTEGER PROGRAMMING. The Theory of Valid Inequalities. Strong Valid Inequalities and Facets for Structured Integer Programs. Duality and Relaxation. General Algorithms. Special-Purpose Algorithms. Applications of Special- Purpose Algorithms. COMBINATORIAL OPTIMIZATION. Integral Polyhedra. Matching. Matroid and Submodular Function Optimization. References. Indexes.