Obtaining Quality-Proved Near Optimal Results for Traveling Salesman Problem

TL;DR

This paper introduces a novel probabilistic approach using Truncated Generalized Beta distribution to achieve near-optimal solutions for the symmetric TSP, surpassing the longstanding Christofides approximation bound.

Contribution

It proposes the first probabilistic model for TSP optimal tour lengths and an iterative method to obtain quality-proved near optimal solutions.

Findings

Approaches the true TSP optimum as iterations increase.

Provides a new probabilistic framework for TSP approximation.

Achieves a near 1.5-approximation bound with iterative improvements.

Abstract

The traveling salesman problem (TSP) is one of the most challenging NP-hard problems. It has widely applications in various disciplines such as physics, biology, computer science and so forth. The best known approximation algorithm for Symmetric TSP (STSP) whose cost matrix satisfies the triangle inequality (called $\triangle$STSP) is Christofides algorithm which was proposed in 1976 and is a $\frac{3}{2}$-approximation. Since then no proved improvement is made and improving upon this bound is a fundamental open question in combinatorial optimization. In this paper, for the first time, we propose Truncated Generalized Beta distribution (TGB) for the probability distribution of optimal tour lengths in a TSP. We then introduce an iterative TGB approach to obtain quality-proved near optimal approximation, i.e., (1+$\frac{1}{2}(\frac{\alpha+1}{\alpha+2})^{K-1}$)-approximation where $K$ is the number of iterations in TGB and $\alpha (>>1)$ is the shape parameters of TGB. The result can approach the true optimum as $K$ increases.