Karp's patching algorithm on random perturbations of dense digraphs

We consider the following question. We are given a dense digraph $D_0$ with minimum in- and out-degree at least $\alpha n$, where $\alpha >0$ is a constant. We then add random edges $R$ to $D_0$ to create a digraph $D$. Here an edge $e$ is placed independently into $R$ with probability $n^{-\epsilon }$ where $\epsilon >0$ is a small positive constant. The edges $E(D)$ of $D$ are given independent edge costs $C=C(e),e\in E(D)$, where $C$ has a density $f(x)=a+bx+o(x)$ as $x\to 0$. Here $a>0,b$ are constants. The prime examples will be the uniform $[0,1]$ distribution ($a=1,b=0$) and the exponential mean 1 distribution $EXP(1)$ ($a=1,b=-1$). 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 whose cost is asymptotically equal to the cost of the associated assignment problem. Karp’s algorithm runs in polynomial time.