Karp's patching algorithm on dense digraphs

TL;DR

This paper analyzes Karp's patching algorithm on dense directed graphs with random edge costs, showing it finds near-optimal tours efficiently in the asymmetric TSP setting.

Contribution

It proves that Karp's patching algorithm yields asymptotically optimal solutions for dense random digraphs with high probability.

Findings

Karp's algorithm finds tours asymptotically equal to the assignment problem.

The algorithm runs in polynomial time.

Results hold for graphs with minimum in- and out-degree greater than half the vertices.

Abstract

We consider the following question. We are given a dense digraph $D$ with minimum in- and out-degree at least $\alpha n$, where $\alpha>1/2$ is a constant. The edges of $D$ are given edge costs $C(e),e\in E(D)$, where $C(e)$ is an independent copy of the uniform $[0,1]$ random variable $U$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E(D)$. We show that w.h.p. the patching algorithm of Karp finds a tour for the asymmetric traveling salesperson problem that is asymptotically equal to that of the associated assignment problem. Karp's algorithm runs in polynomial time.