# An Upper Bound for Minimum True Matches in Graph Isomorphism with Simulated Annealing

## TL;DR

This paper demonstrates that simulated annealing is unlikely to find near-optimal solutions for graph isomorphism, providing theoretical bounds and experimental evidence of its limitations in achieving accurate matches.

## Contribution

It offers a theoretical upper bound on the success of simulated annealing for graph isomorphism and validates this with experimental results showing low probabilities of correct matches.

## Key findings

- Simulated annealing has a very low probability of achieving more than three correct matches.
- Theoretical bounds suggest simulated annealing is ineffective for near-optimal solutions.
- Experimental results confirm the theoretical limitations in practical scenarios.

## Abstract

Graph matching is one of the most important problems in graph theory and combinatorial optimization, with many applications in various domains. Although meta-heuristic algorithms have had good performance on many NP-Hard and NP-Complete problems, for this problem there are not reported superior solutions by these algorithms. The reason of this inefficiency has not been investigated yet. In this paper we show that simulated annealing as an stochastic optimization method is unlikely to be even close to the optimal solution for this problem. In addition to theoretical discussion, the experimental results also verified our idea; for example, in two sample graphs, the probability of reaching to a solution with more than three correct matches is about $0.02$ in simulated annealing.