TSP路径的八分之五近似
Eight-Fifth Approximation for TSP Paths
摘要：
我们证明了近似比8/5的度量$\{s,t\}$ -路径TSP问题，更一般的最短连接$T$ -加入。实现该比率的算法是由An、Kleinberg和Shmoys（2012）提出的Christofides算法（1976）的简单“Best of Many”版本，其包括从具有由分数松弛的最优解$x^*$支配的凸组合的树族$\Fscr_{>0}$构建的那些树中确定最佳Christofides $\{s,t\}$ -tour。他们给出了这种$\{s,t\}$ -环的近似保证$\frac{\sqrt{5}+1}{2}$，这是Hoogeveen的Christofides型算法（1991）的5/3保证之后的第一个改进. Cheriyan，Friggstad and Gao（2012）将该结果扩展到$|T|\ge 4$的最短连接$T$ -连接的13/8近似。通过简化和改进An，Kleinberg和Shmoys的方法证明了8/5的比率，该方法包括完成$x^*/2$以控制生成树的“奇偶校正”成本。我们将$\Fscr_{>0}$中每个生成树的边集划分为一个$\{s,t\}$ -路径（或者更一般地说，划分为一个$T$ -连接）及其补集，从而导致了$x^*$的分解.这种分解可以被细化，然后有效地用于完成$x^*/2$，而不使用线性编程或$T$的特定属性，而是通过向$x^*$中固有的、单独定制的显式给定向量添加对于$x^*/2$有缺陷的每个切割。一个简单的例子表明，Best of Many Christofides算法可能找不到比问题及其分数松弛的偶然常见最优解的3/2倍更短的$\{s,t\}$ -行程。
英文摘要：
We prove the approximation ratio 8/5 for the metric $\{s,t\}$-path-TSP problem, and more generally for shortest connected $T$-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides $\{s,t\}$-tour out of those constructed from a family $\Fscr_{>0}$ of trees having a convex combination dominated by an optimal solution $x^*$ of the fractional relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for such an $\{s,t\}$-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected $T$-joins, for $|T|\ge 4$. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing $x^*/2$ in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in $\Fscr_{>0}$ into an $\{s,t\}$-path (or more generally, into a $T$-join) and its complement, which induces a decomposition of $x^*$. This decomposition can be refined and then efficiently used to complete $x^*/2$ without using linear programming or particular properties of $T$, but by adding to each cut deficient for $x^*/2$ an individually tailored explicitly given vector, inherent in $x^*$. A simple example shows that the Best of Many Christofides algorithm may not find a shorter $\{s,t\}$-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.
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