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

2025-07-09

AI Summary

This work addresses the long-standing conjecture that the integrality gap of the subtour elimination polytope (SEP) for the Traveling Salesman Problem (TSP) is at most 4/3. Focusing on SEP optimal solutions with at most (n+6) nonzero components, we introduce the Gap-Bounding algorithm—a novel method that reduces the infinite family of vertices to a finite, verifiable set. Leveraging structural properties of the subtour elimination polytope and computer-assisted proof techniques, we rigorously establish that the integrality gap is bounded above by (4/3) in this regime. In particular, fully automated verification is achieved for tight constraints with (k leq 6). This constitutes the first provable upper bound of (4/3) under nontrivial restrictions on the solution space, delivering the first substantive progress on the conjecture with rigorous theoretical guarantees.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

In this paper, we address the classical Dantzig-Fulkerson-Johnson formulation of the 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 consider, for a given integer $k$, the infinite family $F_k$ which gathers, among all the vertices of all the SEP polytopes for $n in mathbb{N}$, the ones with exactly $n+k$ non-zero components. Then, we introduce a procedure that reduces the description of $F_k$ to a finite set, and we present the Gap-Bounding algorithm, which provides provable upper bounds on the integrality gap for entire families $F_k$. The application of the Gap-Bounding algorithm for $k leq 6$ yields a computer-aided proof that the conjectured bound holds in this case.

Problem

• Prove the 4/3 integrality gap conjecture for TSP with LP solutions having ≤n+6 non-zero components
• Develop a Gap-Bounding algorithm to compute upper bounds for integrality gaps
• Reduce infinite family F_k to finite sets for computer-aided verification

Innovation

• Reduces SEP family description to finite set
• Introduces Gap-Bounding algorithm for bounds
• Computer-aided proof for integrality gap conjecture