Hard to Solve Instances of the Euclidean Traveling Salesman Problem

TL;DR

This paper introduces a new family of Euclidean TSP instances that demonstrate the integrality ratio approaching 4/3 and are computationally challenging for existing solvers, serving as potential benchmarks.

Contribution

The authors present a novel family of Euclidean TSP instances with proven integrality ratio convergence to 4/3 and high computational difficulty, expanding benchmark options.

Findings

Integrality ratio of the subtour LP converges to 4/3 for the new instances.

Existing solvers like Concorde require days to solve these instances.

New instances are significantly harder than similar-sized TSPLIB instances.

Abstract

The well known $4/3$ conjecture states that the integrality ratio of the subtour LP is at most $4/3$ for metric Traveling Salesman instances. We present a family of Euclidean Traveling Salesman instances for which we prove that the integrality ratio of the subtour LP converges to $4/3$. These instances (using the rounded Euclidean norm) turn out to be hard to solve exactly with Concorde, the fastest existing exact TSP solver. For a 200 vertex instance from our family of Euclidean Traveling Salesman instances Concorde needs several days of CPU time. This is more than 1,000,000 times the runtime for a TSPLIB instance of similar size. Thus our new family of Euclidean Traveling Salesman instances may serve as new benchmark instances for TSP algorithms.