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arXiv 1405.0945cs.DS
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关于Sherali-Adams族中非对称TSP的整比

On Integrality Ratios for Asymmetric TSP in the Sherali-Adams Hierarchy

Joseph Cheriyan, Zhihan Gao, Konstantinos Georgiou, Sahil Singla

备注：
26 pages, 7 figures. An extended abstract of this work appeared in the proceedings of the 40th International Colloquium on Automata, Languages, and Programming ({ICALP} 2013)
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摘要：

我们研究了ATSP（非对称旅行商问题），我们的重点是在Sherali-Adams（SA）电梯和项目方法的框架下的负面结果。我们的主要结果涉及到标准的LP（线性规划）ATSP松弛，由于Dantzig，Fulkerson和约翰逊。对于任意固定整数$t\geq 0$和小$ε$，$0<ε\ll{1}$，存在$ν=ν(t,ε)=O(t/ε)$顶点上的有向图$G$，使得从$G$上的标准LP开始的SA系统的水平$t$的完整性比为$\ge 1+\frac{1-ε}{2t+3} \approx \frac43, \frac65, \frac87, \dots$ .因此，就输入大小而言，结果适用于任何$t = 0,1,\dots,Θ(ν)$级别。我们的主要贡献是确定的有向图的结构属性，使我们能够构建分数可行的解决方案，从标准的~LP开始的SA系统的任何级别~ $t$。我们的硬实例是简单的，并满足结构性质。有一个进一步放松的标准LP称为平衡LP，和我们的方法大大简化时，SA系统的起始LP是平衡的~LP;特别是，相关的结构性质（有向图）简化，使得它是由著名的Charikar，Goemans和Karloff（CGK）的建设给出的有向图满足。因此，CGK有向图作为硬实例，我们得到了$1 +\frac{1-ε}{t+1}$对SA系统的任何水平$t$的完整性比，其中$0<ε\ll{1}$和顶点数为$ν(t,ε)=O((t/ε)^{(t/ε)})$。此外，我们对标准~LP的结果扩展到Path-ATSP（找到一个最小成本的哈密尔顿双路从一个给定的源顶点到一个给定的汇顶点）。

英文摘要：

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer $t\geq 0$ and small $ε$, $0<ε\ll{1}$, there exists a digraph $G$ on $ν=ν(t,ε)=O(t/ε)$ vertices such that the integrality ratio for level~$t$ of the SA system starting with the standard LP on $G$ is $\ge 1+\frac{1-ε}{2t+3} \approx \frac43, \frac65, \frac87, \dots$. Thus, in terms of the input size, the result holds for any $t = 0,1,\dots,Θ(ν)$ levels. Our key contribution is to identify a structural property of digraphs that allows us to construct fractional feasible solutions for any level~$t$ of the SA system starting from the standard~LP. Our hard instances are simple and satisfy the structural property. There is a further relaxation of the standard LP called the balanced LP, and our methods simplify considerably when the starting LP for the SA system is the balanced~LP; in particular, the relevant structural property (of digraphs) simplifies such that it is satisfied by the digraphs given by the well-known construction of Charikar, Goemans and Karloff (CGK). Consequently, the CGK digraphs serve as hard instances, and we obtain an integrality ratio of $1 +\frac{1-ε}{t+1}$ for any level~$t$ of the SA system, where $0<ε\ll{1}$ and the number of vertices is $ν(t,ε)=O((t/ε)^{(t/ε)})$. Also, our results for the standard~LP extend to the Path-ATSP (find a min cost Hamiltonian dipath from a given source vertex to a given sink vertex).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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