Algorithm Design: Given an undirected graph where each node represents a city and each edge represents a road between two cities, design an algorithm to find the shortest route that visits each city exactly once and returns to the origin city. This problem is known as the Traveling Salesman Problem $($TSP$).$ The challenge is to create an efficient algorithm, considering that TSP is NP$-$hard and no polynomial$-$time solution is known for large instances.

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