A Method to Compute the Sparse Graphs for Traveling Salesman Problem Based on Frequency Quadrilaterals

TL;DR

This paper introduces an iterative method to generate sparse graphs for the TSP using frequency quadrilaterals, reducing computation time while preserving the optimal cycle with high probability.

Contribution

The paper presents a novel iterative algorithm that efficiently computes sparse graphs for TSP based on frequency quadrilaterals, improving on existing methods.

Findings

The algorithm reduces the number of edges to O(n log n) while maintaining the optimal cycle.

It lowers computation time to O(N n^2) for most TSP instances.

Experiments confirm the method's effectiveness on TSPLIB instances.

Abstract

In this paper, an iterative algorithm is designed to compute the sparse graphs for traveling salesman problem (TSP) according to the frequency quadrilaterals so that the computation time of the algorithms for TSP will be lowered. At each computation cycle, the algorithm first computes the average frequency \bar{f}(e) of an edge e with N frequency quadrilaterals containing e in the input graph G(V,E). Then the 1/3|E| edges with low frequency are eliminated to generate the output graph with a smaller number of edges. The algorithm can be iterated several times and the original optimal Hamiltonian cycle is preserved with a high probability. The experiments demonstrate the algorithm computes the sparse graphs with the O(nlog_2n) edges containing the original optimal Hamiltonian cycle for most of the TSP instances in the TSPLIB. The computation time of the iterative algorithm is O(Nn^2).