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Graph matching or quadratic assignment, is the problem of labeling thevertices of two graphs so that they are as similar as possible. A common methodfor approximately solving the NP-hard graph matching problem is relaxing it toa convex optimization problem over the set of doubly stochastic (DS) matrices.Recent analysis has shown that for almost all pairs of isomorphic andasymmetric graphs, the DS relaxation succeeds in correctly retrieving theisomorphism between the graphs. Our goal in this paper is to analyze the caseof symmetric isomorphic graphs. This goal is motivated by shape matchingapplications where the graphs of interest usually have reflective symmetry.  For symmetric problems the graph matching problem has multiple isomorphismsand so convex relaxations admit all convex combinations of these isomorphismsas viable solutions. If the convex relaxation does not admit any additionalsuperfluous solution we say that it is convex exact. In this case there aretractable algorithms to retrieve an isomorphism from the convex relaxation.  We show that convex exactness depends strongly on the symmetry group of thegraphs; For a fixed symmetry group $G$, either the DS relaxation will be convexexact for almost all pairs of isomorphic graphs with symmetry group $G$, or theDS relaxation will fail for all such pairs. We show that for reflective groupswith at least one full orbit convex exactness holds almost everywhere, andprovide some simple examples of non-reflective symmetry groups for which convexexactness always fails.  When convex exactness holds, the isomorphisms of the graphs are the extremepoints of the convex solution set. We suggest an efficient algorithm forretrieving an isomorphism in this case. We also show that the "convex toconcave" projection method will also retrieve an isomorphism in this case.

Exact Recovery with Symmetries for the Doubly-Stochastic Relaxation