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.
翻译:我们考虑下面的问题。 我们得到的是一个密集的估算值$D, 最低度和最低度在度和体外至少为$ alpha n$, 其中$\ alpha>1/2 美元是一个常数。 $D的边缘值被给出了边缘成本$C( e), e\ in E( D)$, 其中$C( e) 是制服 $[ 0, 1美元随机变量美元的独立副本。 请用$C( i, j), i, j\in[ n] 美元作为相关的n\ times n$n( i,j) infty$, 如果$( j)\ notin E( D)$的话。 我们显示, Karp 的修补算算法找到了一个与相关分配问题几乎相等的不对称旅行销售人员问题巡回旅行。 Karp 算法在多元时间运行 。