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A Canceling Heuristic for the Directed Traveling Salesman Problem

Steffen Borgwardt, Zachary Sorenson

Abstract

The Traveling Salesman Problem (TSP) is one of the classic and hard problems in combinatorial optimization. We develop a new heuristic that uses a connection between Minimum Cost Flow Problems and the TSP to improve on a given suboptimal tour, such as a local optimum found using a classic heuristic. Minimum Cost Flow Problems can be solved efficiently through linear programming or combinatorial algorithms based on cycle canceling. We investigate the potential of flow-canceling in the context of the TSP. Through a restriction of the search space to cycles and circulations that alternate between arcs in- and outside of the tour, practical results exhibit that only a low number of subtours is created, and a lightweight patching step suffices for a high success rate and gap closure towards an optimum.

A Canceling Heuristic for the Directed Traveling Salesman Problem

Abstract

The Traveling Salesman Problem (TSP) is one of the classic and hard problems in combinatorial optimization. We develop a new heuristic that uses a connection between Minimum Cost Flow Problems and the TSP to improve on a given suboptimal tour, such as a local optimum found using a classic heuristic. Minimum Cost Flow Problems can be solved efficiently through linear programming or combinatorial algorithms based on cycle canceling. We investigate the potential of flow-canceling in the context of the TSP. Through a restriction of the search space to cycles and circulations that alternate between arcs in- and outside of the tour, practical results exhibit that only a low number of subtours is created, and a lightweight patching step suffices for a high success rate and gap closure towards an optimum.

Paper Structure

This paper contains 16 sections, 5 theorems, 20 equations, 7 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions and Outline
  4. The Cancel Step
  5. The Search Space
  6. Computational Strategy
  7. Implementation
  8. Variants of the Cycap Heuristic
  9. Tailored Versions of Classic Algorithms
  10. Computational Experiments
  11. Success Rate
  12. Gap Closure
  13. Running Time
  14. Cycap-only times
  15. Embedded runs with $k$-opt
  16. ...and 1 more sections

Key Result

Theorem 1

Let $R=(G,T)$ be a tour graph and let $A$ be a directed cycle in $R$. Then

Figures (7)

  • Figure 1: Flow Chart for the Cycap Heuristic
  • Figure 2: A non-simple tour-alternating cycle. Insertion arcs are blue, removal arcs are red.
  • Figure 3: An example in which a cycle cancel outperforms $2$-opt and $3$-opt. a) Locally optimal tour for $2$-opt and $3$-opt b) Negative cycle in tour graph c) Improved tour formed by cancel
  • Figure 4: An example with opposite non-tour arcs. a) Initial tour $T$ b) Tour-alternating cycle $A$ c) Tour-alternating cycle $A'$ d) Tour-alternating circulation $C$ composed of $A$ and $A'$ e) Flow formed by $C$ with a trim of opposite non-tour arcs
  • Figure 5: Separated graph construction. a) Tour graph $R=(G,T)$. b) Separated graph $R'$.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 1: Tour graph
  • Definition 2: Tour–alternating cycle
  • Definition 3: Tour–alternating circulation
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Definition 4: Separated graph
  • ...and 3 more