A Non-Degenerate Perturbation of the Assignment Polytope and its Application to Graph Matching

TL;DR

This paper introduces a simple perturbation method to transform the assignment polytope into a non-degenerate form, enabling convex maximization approaches to effectively solve graph matching problems, which were previously challenging due to degeneracy issues.

Contribution

The paper presents a perturbation scheme that creates a non-degenerate surrogate problem equivalent to the original, specifically applied to graph matching, addressing an open issue in convex optimization approaches.

Findings

Surrogate problem is non-degenerate and equivalent to the original

Method applies to graph matching as a convex maximization problem

Resolves open issue in convex optimization for graph matching

Abstract

We consider maximizing a continuous convex function over the assignment polytope. Such problems arise in Graph Matching (the optimization version of Graph Isomorphism) and Quadratic Assignment problems. In the typical case of maximizing a convex function over a polytope the problem can be solved by using a simplicial algorithm such as Tuy's method or the Falk-Hoffman method, but these algorithms require that the underlying polytope be non-degenerate, which is not the case for the assignment polytope. In this note we show how a simple perturbation scheme can be used to create a "surrogate problem" that is both non-degenerate and combinatorially equivalent to the original problem. We further provide an explicit construction of a surrogate problem that is non-degenerate and combinatorially equivalent to the Graph Matching problem, when the latter is posed as a convex maximization problem. By constructing a surrogate problem that is known a priori to be non-degenerate and combinatorially equivalent to Graph Matching we resolve an "open issue" of solving Graph Matching via convex maximization first raised by Maciel.