Karp's patching algorithm on dense digraph

TL;DR

This paper analyzes Karp's patching algorithm on dense directed graphs with random edge costs, showing it efficiently finds near-optimal tours for the asymmetric TSP under certain probabilistic cost models.

Contribution

The paper demonstrates that a modified version of Karp's patching algorithm can find asymptotically optimal tours in dense digraphs with random costs, running in polynomial time.

Findings

Algorithm finds tours asymptotically equal to the assignment problem

Runs in polynomial time with high probability

Applicable under specific probabilistic cost distributions

Abstract

We consider the following question. We are given a dense digraph $D$ with $n$ vertices and minimum in- and out-degree at least $\alpha n$, where $\alpha>1/2$ is a constant. The edges $E(D)$ of $D$ are given independent edge costs $C(e),e\in E(D)$, such that (i) $C$ has a density $f$ that satisfies $f(x)=a+bx+O(x^2)$, for constants $a>0,b$ as $x\to 0$ and such that in general either (ii) $\Pr(C\geq x)\leq \a e^{-\b x}$ for constants $\a,\b>0$, or $f(x)=0$ for $x>\n$ for some constant $\n>0$. Let $C(i,j),i,j\in[n]$ be the associated $n\times n$ cost matrix where $C(i,j)=\infty$ if $(i,j)\notin E$. We show that w.h.p. (a small modification to) 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. The algorithm runs in polynomial time.