Solution Attractor of Local Search System: A Method to Reduce Computational Complexity of the Traveling Salesman Problem

Weiqi Li

doi:10.5772/intechopen.90521

Abstract

The traveling salesman problem (TSP) is presumably difficult to solve exactly using local search algorithms. It can be exactly solved by only one algorithm—the enumerative search algorithm. However, the scanning of all possible solutions requires exponential computing time. Do we need exploring all the possibilities to find the optimal solution? How can we narrow down the search space effectively and efficiently for an exhausted search? This chapter attempts to answer these questions. A local search algorithm is a discrete dynamical system, in which a search trajectory searches a part of the solution space and stops at a locally optimal point. A solution attractor of a local search system for the TSP is defined as a subset of the solution space that contains all locally optimal tours. The solution attractor concept gives us great insight into the computational complexity of the TSP. If we know where the solution attractor is located in the solution space, we simply completely search the solution attractor, rather than the entire solution space, to find the globally optimal tour. This chapter describes the solution attractor of local search system for the TSP and then presents a novel search system—the attractor-based search system—that can solve the TSP much efficiently with global optimality guarantee.

Highlights

What it is that makes the traveling salesman problem (TSP) difficulty? The difficulty of the TSP is associated with the combinatorial explosion of potential solutions in the solution space
In the attractor-based search system (ABSS), K search trajectories start a sample of initial tours from uniform distribution over the solution space and, through a randomized local search process, generate a sample of locally optimal tours that are uniformly distributed in the constructed solution attractor
For the TSP, the computational complexity is associated with the combinatorial explosion of potential solutions in the solution space

Summary

Introduction

What it is that makes the TSP difficulty? The difficulty of the TSP is associated with the combinatorial explosion of potential solutions in the solution space. Numerous approaches to solving the TSP have been published Some algorithms such as enumerative search, branch-and-bound search, and linear programming are exact approaches but lack efficiency. The scope of a single search trajectory is limited by the neighborhood definition Both the TSP and local search have been hot research topics for decades, and many aspects of them have been studied. If we can quickly identify that small region, the solution attractor, and search that region thoroughly in reasonable time, the computational complexity of the problem can be dramatically reduced or may not exist. This chapter introduces the solution attractor concept, which helps us understand the behavior of a local search system for the TSP and offers an important method to solve the problem efficiently with global optimality guarantee. We present a novel search algorithm—the attractor-based search system (ABSS)—that is a simple and quick global search system for the TSP

Reframing the TSP definition

Solution attractor of local search system for TSP

Global optimization and computational complexity of ABSS

How does the ABSS meet the requirements of a global optimization system?

Findings

Conclusions