A Unified Framework for Integer Programming Formulation of Graph Matching Problems

Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program then solved using exact, heuristic and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding of complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The aim of the present study is to provide a unified framework for IP formulation of graph matching problems. Although there are many surveys on graph matching problems, however, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of difficult problems arising in practice, especially in pattern analysis

Evaluating a Clique Partitioning Problem Model for Clustering High-Dimensional Data Mining

This paper considers the problem of clustering high dimensional data as a clique partitioning problem. Data objects within a cluster have high degree of similarity. The similarity index values are first constructed into a graph as a clique partitioning problem which can be formulated into a form of unconstrained quadratic program model and then solved by a tabu search heuristic incorporating strategic oscillation with a critical event memory. Results from other clustering techniques are compared on a set of instances from open literatures. The computational results highlight the robustness of this new model and solution methodology

Inventory control, by Axsater, S. Boston: Kluwer Academic Publishers, 2000, 202 pp., $95.00 (cloth).

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Unrelated Parallel Machine Selection and Job Scheduling with the Objective of Minimizing Total Workload and Machine Fixed Costs

This paper is concerned with scheduling of a set of n single-operation tasks/orders on a set of m unrelated parallel machines where subcontracting is allowed. When a machine/subcontractor is chosen to do a set of orders/tasks, it incurs a one-time fixed cost. When a job/order is performed by a machine/subcontractor, there is a cost that depends on the machine/subcontractor. The objective is to choose a subset of k machines and/or subcontractors from the set of all m available machines and/or subcontractors to perform all jobs to minimize the sum of total workload costs and total fixed costs. We discuss the complexity of the problem, and prove NP-hardness of the problem. Simplified mathematical development is provided that allows efficient implementation of two-exchange algorithms. An efficient tabu search heuristic with a diversification generation component is developed. An extensive computational experiment of the heuristic for large-scale problems with comparison to the results from CPLEX software is presented. We also solved 40 benchmark k-median problems available on the Internet that have been used by many researchers. Note to Practitioners - To be competitive in the global market, companies must be prudent in the use of their resources. This paper considers a parallel scheduling environment where choosing in-house machines and/or subcontractors as resources to perform the orders/jobs is the main objective. Processing time (or cost) of a job to be performed by different machines or subcontractors can be different. Furthermore, if a machine or a subcontractor is chosen to perform a set of orders, there is a one-time fixed cost (in the case of subcontractor it can be considered transportation cost) that depends on the machine or subcontractor. The scheduling criteria are to choose a subset of k machines and/or subcontractors to do all orders/jobs while minimizing the sum of the total workload and total fixed costs. The complexity of the problem is discussed and shown to be NP-hard. An efficient tabu search that solves large-scale problems in fraction of a second of CPU time is presented and an extensive computational experiment is provided

On zero duality gap in surrogate constraint optimization: The case of rational-valued functions of constraints

This paper is concerned with the constrained optimization problem. A detailed discussion of surrogate constraints with zero duality gaps is presented. Readily available surrogate multipliers are considered that close the duality gaps where constraints are rational-valued. Through illustrative examples, the sources of duality gaps are examined in detail. While in the published literature, in many situations conclusions have been made about the existence of non-zero duality gaps, we show that taking advantage of full problem information can close the duality gaps. Overlooking such information can produce shortcomings in the research in which a non-zero duality gap is observed. We propose theorems to address the shortcomings and report results regarding implementation issues. © 2011 Elsevier Inc