Abstract
Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the Θ(n3) enumeration of all triples is likely to exist for this problem, but algorithms with average case O(n3−ϵ) are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by O(n2.5)) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances.
Highlights
The Traveling Salesman Problem (TSP), in all likelihood the most famous combinatorial optimization problem, calls for finding the shortest Hamiltonian cycle in a complete graph G = (V, E)of n nodes, weighted on the arcs
The experiments were run on two types of graphs: (i) random graphs generated by us and (ii) instances from the standard benchmark repository Traveling Salesman Problem Library (TSPLIB) [16]
We have described an algorithmic strategy for optimizing the 3-OPT neighborhood in an effective way
Summary
Of n nodes, weighted on the arcs. In this paper we consider the symmetric TSP, i.e., the graph is undirected and the distance between two nodes is the same irrespective of the direction in which we traverse an edge. Let us denote by c(i, j) = c( j, i ) the distance between any two nodes i and j. We call each solution of the problem a tour. A tour is identified by a permutation of vertices N − 1, and {vn , v1 } the edges of the tour. The length of a tour T, denoted by c( T ) is the sum of the lengths of the edges of the tour. For any set F of edges, we denote by c( F ) the value ∑e∈ F c(e)
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