Theory of Complexity in Social Systems

**Chapter 1**

*Weiqi Li*

**Abstract**

dynamical systems, attractor

**1. Introduction**

**3**

How to Solve the Traveling

The Traveling Salesman Problem (TSP) is believed to be an intractable problem and have no practically efficient algorithm to solve it. The intrinsic difficulty of the TSP is associated with the combinatorial explosion of potential solutions in the solution space. When a TSP instance is large, the number of possible solutions in the solution space is so large as to forbid an exhaustive search for the optimal solutions. The seemingly "limitless" increase of computational power will not resolve its genuine intractability. Do we need to explore all the possibilities in the solution space to find the optimal solutions? This chapter offers a novel perspective trying to overcome the combinatorial complexity of the TSP. When we design an algorithm to solve an optimization problem, we usually ask the critical question: "How can we find all exact optimal solutions and how do we know that they are optimal in the solution space?" This chapter introduces the Attractor-Based Search System (ABSS) that is specifically designed for the TSP. This chapter explains how the ABSS answer this critical question. The computing complexity of the ABSS is also discussed.

**Keywords:** combinatorial optimization, global optimization, heuristic local search, computational complexity, traveling salesman problem, multimodal optimization,

The TSP is one of the most intensively investigated optimization problems and often treated as the prototypical combinatorial optimization problem that has provided much motivation for design of new search algorithms, development of complexity theory, and analysis of solution space and search space [1, 2]. The TSP is defined as a complete graph *Q* ¼ ð Þ *V*, *E*,*C* , where *V* ¼ *vi* f g : *i* ¼ 1, 2, … , *n* is a set of *<sup>n</sup>* nodes, *<sup>E</sup>* <sup>¼</sup> f g *e i*ð Þ , *<sup>j</sup>* : *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>*; *<sup>i</sup>* 6¼ *<sup>j</sup> <sup>n</sup>*�*<sup>n</sup>* is an edge matrix containing the set of edges that connects the *<sup>n</sup>* nodes, and *<sup>C</sup>* <sup>¼</sup> f g *c i*ð Þ , *<sup>j</sup>* : *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>*; *<sup>i</sup>* 6¼ *<sup>j</sup> <sup>n</sup>*�*<sup>n</sup>* is a cost matrix holding a set of traveling costs associated with the set of edges. The solution space *S* contains a finite set of all feasible tours that a salesman may traverse. A tour *s*∈*S* is a closed route that visits every node exactly once and returns to the starting node at the end. Like many real-world optimization problems, the TSP is inherently multimodal; that is, it may contain multiple optimal tours in its solution space. We assume that a TSP instance *Q* contains *h* ð Þ ≥ 1 optimal tours in *S*. We denote *<sup>f</sup>*(*s*) as the objective function, *<sup>s</sup>* <sup>∗</sup> <sup>¼</sup> min *<sup>s</sup>*∈*Sf s*ð Þ as an optimal tour and *<sup>S</sup>* <sup>∗</sup> as the set of *h* optimal tours. The objective of the TSP is to find all *h* optimal tours in

the solution space, that is, *S* <sup>∗</sup> ⊂*S*. Therefore, the argument is

Salesman Problem

### **Chapter 1**
