Optimization approach in quadratic assignment and graph isomorphism

Abstract

Graph matching problem, also known as quadratic assignment problem (QAP) in a practical context, aims at finding the best correspondence between two given graphs. If two nodes are connected in one graph, it would be ideal if their corresponding images are connected in the other. The optimal solution should maximize the number of matches between the two graphs, while minimizing the weight of mismatches. Graph isomorphism is a particular case where an exact correspondence with no mismatches could be found. The quadratic assignment problem itself is hard to solve due to its high computational complexity. In this paper, the researcher analyzes two optimization approaches in detail: convex relaxation and the Fast Approximate QAP algorithm, both trying to solve the problem with lower computational effort. The performances of these two methods are further compared through numerical experiments. This paper focuses on pairs of equal-sized undirected graphs. ρ correlated graphs are constructed to simulate the noise in inexact graph matching problems. After comparing two existing methods, we propose an even faster optimization approach with comparable accuracy. This new approach provides a particular way to approximate an ideal initialization for the traditional Fast Approximate QAP algorithm. The significant reduction in the runtime makes obtaining good correspondence between large graphs feasible.