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The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components

Tullio Villa, Eleonora Vercesi, Janos Barta, Monaldo Mastrolilli

TL;DR

This work analyzes the integrality gap of the symmetric metric TSP under the Dantzig–Fulkerson–Johnson (DFJ) subtour-elimination relaxation by focusing on SEP vertices with limited support. It introduces the Gap-Bounding algorithm and the vertex-family framework (including ancestors $\\mathcal{A}_k$ and the sets $\\mathcal{F}_k$) to propagate gap bounds from a small, finite set of base cases to infinite families of vertices, enabling a universal bound for costs whose SEP solution lies in these families. By combining duality-based bounds with a constructive bb-move mechanism, the authors prove that the integrality gap is at most $4/3$ for all SEP vertices with at most $n+6$ non-zero components, supported by computational verification up to $k=6$ and augmented by selective successor analysis. The methodology offers a novel, hybrid theoretical-computational path for tackling the long-standing $4/3$ conjecture and may extend to broader classes of SEP vertices and related combinatorial gaps.

Abstract

We address the classical Dantzig - Fulkerson - Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution has at most n + 6 non-zero components, then the conjecture is true. To establish this result, we devise a new methodology that combines theoretical analysis and computational verification.

The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components

TL;DR

This work analyzes the integrality gap of the symmetric metric TSP under the Dantzig–Fulkerson–Johnson (DFJ) subtour-elimination relaxation by focusing on SEP vertices with limited support. It introduces the Gap-Bounding algorithm and the vertex-family framework (including ancestors and the sets ) to propagate gap bounds from a small, finite set of base cases to infinite families of vertices, enabling a universal bound for costs whose SEP solution lies in these families. By combining duality-based bounds with a constructive bb-move mechanism, the authors prove that the integrality gap is at most for all SEP vertices with at most non-zero components, supported by computational verification up to and augmented by selective successor analysis. The methodology offers a novel, hybrid theoretical-computational path for tackling the long-standing conjecture and may extend to broader classes of SEP vertices and related combinatorial gaps.

Abstract

We address the classical Dantzig - Fulkerson - Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution has at most n + 6 non-zero components, then the conjecture is true. To establish this result, we devise a new methodology that combines theoretical analysis and computational verification.

Paper Structure

This paper contains 15 sections, 12 theorems, 21 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution and outlook.
  3. Outline.
  4. Background material
  5. Preliminaries from graph theory
  6. Preliminaries from the literature on the integrality gap
  7. Families of vertices with a given number of edges
  8. Redefining the Gap problem
  9. The Gap-Bounding algorithm
  10. Dual formulations of the OPT+ problem
  11. Bounding the Gap on successors
  12. Computational results
  13. Conclusion
  14. Auxiliary proofs
  15. Proof of Lemma

Key Result

theorem thmcountertheorem

Let $\bm{x} \in P_{\text{SEP}}^{n}$ be a vertex. Then $|E_{\bm{x}}| \leq 2n - 3$.

Figures (6)

  • Figure 1: Example of a vertex considered in this work but not in previous literature. Dashed edges may be replaced by arbitrarily long paths of edges of weight $1$.
  • Figure 2: Vertices of $\mathcal{A}_4$, up to isomorphism.
  • Figure 3: Flowchart of the pipeline applied to bound Gap for a successor $\bm{x}'$ of $\bm{x}_0$.
  • Figure 4: Variables $\bm{\mu}^2$ for $\mathcal{D}\,\text{^}\text{II}\text{OPT}\textsuperscript{II}(\bm{x}_2)$ constructed from variables $\bm{\mu}^0$ for $\mathcal{D}\,\text{^}\text{II}\text{OPT}\textsuperscript{II}(\bm{x}_0)$.
  • Figure 5: Correlation between the number of edges in the support graph of a vertex and their integrality gap. Each plot collects all the vertices of $P_{\text{SEP}}^{n}$ for a fixed $n$: $n = 10$ on the left, $n = 11$ on the right. The observed trend is the same for all small $n \leq 12$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • definition thmcounterdefinition: Hamiltonian walk, Hamiltonian tour
  • remark thmcounterremark
  • theorem thmcountertheorem: art:BenBoy:IGSmallTSPart:BoyPul:SEPPoly
  • definition thmcounterdefinition: $1$-edge, $1$-path, from art:BenBoy:IGSmallTSP
  • theorem thmcountertheorem: art:BenBoy:IGSmallTSP
  • definition thmcounterdefinition: bb-move, from art:BenBoy:IGSmallTSP
  • theorem thmcountertheorem: art:BenBoy:IGSmallTSP
  • lemma thmcounterlemma
  • definition thmcounterdefinition: Ancestor of order $k$
  • definition thmcounterdefinition: Successor
  • ...and 17 more