A Necessary and Sufficient Condition for Graph Matching Being Equivalent to the Maximum Weight Clique Problem

TL;DR

This paper establishes a necessary and sufficient condition under which graph matching problems are equivalent to maximum weight clique problems, enabling unified solutions and broad applicability.

Contribution

It introduces a general condition that characterizes when graph matching can be reformulated as a maximum weight clique problem, applicable to many practical scenarios.

Findings

The condition is broad enough to cover many graph matching problems.

Proving equivalence reduces to verifying the condition.

Enables continuous solution approaches for graph matching.

Abstract

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition; and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance.