Abstract: 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.

Keywords: Combinatorial optimization, graph matching, integer programming, quadratic assignment problem.

Abstract: There are numerous real life examples done by many people which show that the alternatives of a decision sometimes can reverse their original rank order when new alternatives are added or old ones deleted and without bringing in new criteria. There is no mathematical theorem which proves that rank must always be preserved and there cannot be because of real life and hypothetical counter examples in decision making methods. Rank preservation came to be accepted as the standard because of techniques that could only rate alternatives one at a time treating them as independent. Thus an alternative receives a score and it will not change when other alternatives are added or deleted. All methods that only rate alternatives one at a time, thus always preserving rank, may not lead to the right decision; even if they may be right in certain areas of application. In reality, to determine how good an alternative is on an intangible criterion needs experience and knowledge about other alternatives and hence in their evaluation, the alternatives cannot be completely considered as independent of one another.

Keywords: Decision Making, priorities, ranking, rank preservation, rank reversal.