Scalefree hardness of average-case Euclidean TSP approximation

TL;DR

This paper demonstrates that, assuming P≠NP, many practical Euclidean TSP heuristics cannot approximate the optimal tour length asymptotically, leading to exponential-time branch-and-bound algorithms even on random instances.

Contribution

It establishes the scalefree hardness of average-case Euclidean TSP approximation for a broad class of heuristics, a previously unknown result.

Findings

Most heuristics fail to approximate TSP length asymptotically.

Branch-and-bound algorithms using these heuristics require exponential time.

The results hold even for random Euclidean instances.

Abstract

We show that if P$\neq$NP, then a wide class of TSP heuristics fail to approximate the length of the TSP to asymptotic optimality, even for random Euclidean instances. Previously, this result was not even known for any heuristics (greedy, etc) used in practice. As an application, we show that when using a heuristic from this class, a natural class of branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set), regardless of the particular branching strategy or lower-bound algorithm used.