[Paper Review] Approximating ATSP by Relaxing Connectivity

Ola Svensson|arXiv (Cornell University)|Feb 6, 2015

Complexity and Algorithms in Graphs4 citations

TL;DR

This paper presents a novel approach to approximating the asymmetric traveling salesman problem (ATSP) by relaxing global connectivity requirements into local connectivity conditions, introducing a new problem called Local-Connectivity ATSP. The authors design a 3-light algorithm for this relaxed problem on node-weighted shortest path metrics, and via a generic reduction, achieve a constant-factor approximation for Node-Weighted ATSP, resolving a long-standing open problem in this special case.

ABSTRACT

The standard LP relaxation of the asymmetric traveling salesman problem has been conjectured to have a constant integrality gap in the metric case. We prove this conjecture when restricted to shortest path metrics of node-weighted digraphs. Our arguments are constructive and give a constant factor approximation algorithm for these metrics. We remark that the considered case is more general than the directed analog of the special case of the symmetric traveling salesman problem for which there were recent improvements on Christofides' algorithm. The main idea of our approach is to first consider an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. For this relaxed problem, it is quite easy to give an algorithm with a guarantee of 3 on node-weighted shortest path metrics. More surprisingly, we then show that any algorithm (irrespective of the metric) for the relaxed problem can be turned into an algorithm for the asymmetric traveling salesman problem by only losing a small constant factor in the performance guarantee. This leaves open the intriguing task of designing a "good" algorithm for the relaxed problem on general metrics.

Motivation & Objective

• To address the longstanding open problem of designing a constant-factor approximation algorithm for the asymmetric TSP in node-weighted shortest path metrics.
• To overcome the limitations of existing approaches that rely on cycle covers or LP relaxations with unbounded integrality gaps.
• To introduce a new relaxation of ATSP—Local-Connectivity ATSP—by replacing global connectivity with local connectivity constraints.
• To establish a generic reduction that transforms any constant-light algorithm for Local-Connectivity ATSP into a constant-factor approximation for general ATSP.
• To explore whether a constant-light algorithm exists for Local-Connectivity ATSP on general metrics, which would yield a constant-factor approximation for general ATSP.

Proposed method

• Introduce Local-Connectivity ATSP, a relaxation of ATSP where global connectivity is replaced by local connectivity conditions on vertex subsets.
• Define a 'light' algorithm as one whose solution weight is bounded by a constant factor times a lower bound function lb(S) on subsets S.
• Propose a 3-light algorithm for Local-Connectivity ATSP on node-weighted digraphs, using a primal-dual style construction with iterative cycle cover selection.
• Use a generic reduction: any α-light algorithm for Local-Connectivity ATSP on a given metric can be converted into an O(α)-approximation for general ATSP.
• Leverage the structure of node-weighted metrics, where cycle weights are bounded by the lower bound function, to tighten the analysis and achieve an integrality gap of at most 13 for the Held-Karp relaxation.
• Apply a charging argument and lower bounding technique using the lower bound function lb to prove that the solution weight is within a constant factor of the optimal.

Experimental results

Research questions

• RQ1 Can a constant-factor approximation be achieved for the asymmetric TSP in node-weighted shortest path metrics, where previous methods failed due to unbounded integrality gaps?
• RQ2 Is there a constant-light algorithm for Local-Connectivity ATSP on general metrics, which would yield a constant-factor approximation for general ATSP?
• RQ3 Can the generic reduction from Local-Connectivity ATSP to ATSP be used to transfer approximation guarantees across different metric classes?
• RQ4 What is the tightest possible integrality gap for the Held-Karp relaxation in the node-weighted ATSP setting?
• RQ5 Can the recent progress on STSP for unweighted graphs be extended to node-weighted graphs, yielding a (1.5−ε)-approximation?

Key findings

• A 3-light algorithm is designed for Local-Connectivity ATSP on node-weighted shortest path metrics, proving that such a solution exists with a constant approximation factor.
• The generic reduction shows that any α-light algorithm for Local-Connectivity ATSP yields an O(α)-approximation for general ATSP, enabling transfer of guarantees.
• For node-weighted metrics, the integrality gap of the Held-Karp relaxation is bounded by 13, improving upon previous bounds.
• The approach resolves the open problem of constant-factor approximation for Node-Weighted ATSP, a special case more general than the recent STSP improvements on unweighted graphs.
• The paper leaves open the existence of a constant-light algorithm for Local-Connectivity ATSP on general metrics, which would imply a constant-factor approximation for general ATSP.
• The method provides a new framework for ATSP approximation by focusing on local connectivity, suggesting a path toward resolving the broader ATSP conjecture.