Karp's patching algorithm on dense digraphs

We consider the following question. We are given a dense digraph $D$ with minimum in- and out-degree at least $\alpha n$, where $\alpha >1\mathrm{/}2$ 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 $C(i,j)=\mathrm{\infty }$ if $(i,j)\mathrm{\notin }E(D)$. 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.