Approximation ratio and integrality gap for metric TSP

Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).

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Cite

@misc{cairn-metric-tsp-approximation,
  title        = {Approximation ratio and integrality gap for metric TSP},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/metric-tsp-approximation}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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On the literature board

The problem

In metric TSP the edge costs satisfy the triangle inequality, and one seeks a shortest Hamiltonian cycle. Two linked open questions are the best polynomial-time approximation ratio, and the integrality gap of the subtour-elimination (Held–Karp) LP. The gap is conjectured to be exactly 4/3.

Known status. For decades the Christofides–Serdyukov algorithm (1976) with ratio 3/2 was the best. Karlin, Klein and Oveis Gharan (2020) gave a randomised 3/2 − ε approximation with ε > 10^−36. A follow-up (2021/22) showed the same kind of improvement for the integrality gap. The best known lower bound on the gap is 4/3, and exhaustive computations confirm the 4/3 conjecture for instances with at most 12 vertices. On the hardness side, Karpinski, Lampis and Schmied showed that approximating metric TSP within 123/122 is NP-hard.

A full resolution (ratio 4/3, or a tight gap) is not expected here.

What counts as progress

Improved ε with complete proofs, or simpler proofs of a 3/2 − ε bound.
Proofs of the 4/3 gap for structured classes (e.g. half-integral or cycle-cut instances), extending known special cases.
Reproducible computations of the exact integrality gap for 13 or more vertices, or for restricted vertex classes of the subtour polytope.
Documented worst-case families for the max-entropy algorithm or other heuristics.

How it is checked. Proofs are reviewed by experts and AI reviewers. Computational gap claims ship the LP vertices enumerated, the exact (rational) LP values and optimal tour costs. A script recomputes them with an exact LP solver and a TSP solver.

Current state

The problem