Abstract
In this work a convex relaxation of a subgraph isomorphism problem is proposed, which leads to a new lower bound that can provide a proof that a subgraph isomorphism between two graphs can not be found. The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail. We consider subgraph isomorphism problem instances of simple graphs which means that only the structural information of the two graphs is exploited and other information that might be available (e.g., node positions) is ignored. The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation. Therefore, for problem instances with a lower bound that is larger than zero this represents a proof that a subgraph isomorphism can not exist. But note that conversely, a negative lower bound does not imply that a subgraph isomorphism must be present and only indicates that a subgraph isomorphism can not be excluded. In addition, the relation of our approach and the reformulation of the largest common subgraph problem into a maximum clique problem is discussed.
Highlights
The subgraph isomorphism problem is a well-known combinatorial optimization problem and often involves the problem of finding the appropriate matching too
The main contribution of this paper lies in the convex relaxation of a subgraph isomorphism problem and the identification of a lower bound for this optimization problem
After providing the notation we use, we introduce a combinatorial quadratic optimization formulation for the subgraph isomorphism problem that can be interpreted as an errorcorrecting graph matching approach
Summary
The subgraph isomorphism problem is a well-known combinatorial optimization problem and often involves the problem of finding the appropriate matching too. It is of particular interest in computer vision where it can be exploited to recognise objects. Error-correcting graph matching 5 — known as error-tolerant graph matching—is a general approach to calculate an assignment between the nodes of two graphs. It is based on the minimisation of graph edit costs which result from some predefined edit operations when one graph is turned exactly into the other. It was applied to several problems in the field of computer vision including segmentation/partitioning, grouping, restoration matching , graph seriation , and camera calibration
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