Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Cloven Traveling Salesman: Cycle Covers and the Integrality Gap of Small ATSP Instances

Alessandro Sosso, Ambrogio Maria Bernardelli, Stefano Gualandi

TL;DR

This work proposes a novel enumeration algorithm for computing the integrality gap of small instances of the subtour elimination formulation for the Asymmetric Traveling Salesman Problem, and compute for the first time the exact integrality gap of half-integer vertices of the asymmetric subtour elimination polytope.

Abstract

This work proposes a novel enumeration algorithm for computing the integrality gap of small instances of the subtour elimination formulation for the Asymmetric Traveling Salesman Problem (ATSP). The core idea is to enumerate pairs of vertex-disjoint cycle covers that can be filtered and mapped to half-integer vertices of the subtour elimination polytope. The two-cycle covers are encoded as lexicographically ordered partitions of $n$ numbers, with an encoding that prevents the generation of several isomorphic vertices. However, since not every cycle cover pair can be mapped to a vertex of the subtour elimination polytope, we have designed an efficient property-checking procedure to control whether a given point is a vertex of the asymmetric subtour elimination polytope. The proposed approach turns upside down the algorithms presented in the literature that first generate every possible vertex and later filter isomorphic vertices. With our approach, we can replicate state-of-the-art results for n<=9 in a tiny fraction of time, and we compute for the first time the exact integrality gap of half-integer vertices of the asymmetric subtour elimination polytope for n=10, 11, 12.

The Cloven Traveling Salesman: Cycle Covers and the Integrality Gap of Small ATSP Instances

TL;DR

This work proposes a novel enumeration algorithm for computing the integrality gap of small instances of the subtour elimination formulation for the Asymmetric Traveling Salesman Problem, and compute for the first time the exact integrality gap of half-integer vertices of the asymmetric subtour elimination polytope.

Abstract

This work proposes a novel enumeration algorithm for computing the integrality gap of small instances of the subtour elimination formulation for the Asymmetric Traveling Salesman Problem (ATSP). The core idea is to enumerate pairs of vertex-disjoint cycle covers that can be filtered and mapped to half-integer vertices of the subtour elimination polytope. The two-cycle covers are encoded as lexicographically ordered partitions of numbers, with an encoding that prevents the generation of several isomorphic vertices. However, since not every cycle cover pair can be mapped to a vertex of the subtour elimination polytope, we have designed an efficient property-checking procedure to control whether a given point is a vertex of the asymmetric subtour elimination polytope. The proposed approach turns upside down the algorithms presented in the literature that first generate every possible vertex and later filter isomorphic vertices. With our approach, we can replicate state-of-the-art results for n<=9 in a tiny fraction of time, and we compute for the first time the exact integrality gap of half-integer vertices of the asymmetric subtour elimination polytope for n=10, 11, 12.

Paper Structure

This paper contains 14 sections, 7 theorems, 25 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Gap problem and vertex characterization
  3. The gap problem
  4. Structure of half-integer solutions
  5. Vertex generation
  6. Encoding of cycle-covers
  7. Feasibility and extremality checking
  8. Circuit Extremality Algorithm
  9. Computational results
  10. Conclusions and future work
  11. Algorithms
  12. Canonical Form of a Cover-set Encoding
  13. Omitted Proofs
  14. Highest integrality gap instances

Key Result

Lemma 1

Let $\overline{\bm{x}}\in\mathfrak{X}^{n}_{2}$ and let $\cover[1]\in\mathfrak{C}^{n}$ be the characteristic vector of a cycle cover of the graph associated with $\overline{\bm{x}}$. If $\cover[2] \coloneqq 2\overline{\bm{x}}-\cover[1]$, then $\cover[2]\in\mathfrak{C}^{n}$.

Figures (8)

  • Figure 1: Overview of the proposed approach for computing the maximum integrality gap of half-integer vertices, with references to the results presented in this paper.
  • Figure 2: A solution $\overline{\bm{x}}=\frac{1}{2}\cover[1]+\frac{1}{2}\cover[2]$ that is not an extreme point of $\mathfrak{P}^{n}$, since for all $S\in\mathcal{S}^{n}$ we have $\sum_{uv\in\outdelta\left(S\right)}x_{uv}>1$.
  • Figure 3: Example of a node cycle cover with the associated encoding. Referring to Definition \ref{['def:cover-encoding']}, we have $n=9$, $N = 3$, $p_1 = 4, p_2 = 3, p_3 = 2$.
  • Figure 4: Example cover-set with its associated encoding (in standard form).
  • Figure 5: Examples of circuit partitions for $A=\mathord{\mathrm{Deg}}_{\overline{\bm{x}}}$ on the matrix representations of the solution's characteristic vectors.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1: Half-integer and pure half-integer extreme points
  • Lemma 1
  • proof
  • Theorem 1: Hall's Marriage Theorem
  • Theorem 2: Structure of half-integer vertices
  • proof
  • Remark 1
  • Definition 2: Cover encoding
  • Definition 3: Cover-set encoding
  • Definition 4: Link, circuit, circuit partition
  • ...and 9 more