通过基于长码的归约改进Steiner树的不可近似性结果
Improved Inapproximability Results for Steiner Tree via Long Code Based Reductions
AI总结：
本文研究Steiner树近似问题，通过基于长码的归约，证明除非P = NP，不存在多项式时间算法以优于19/18的常数比近似Steiner树，还关联到唯一博弈猜想，在准二分图特殊情况也有改进，主要贡献是适配集合覆盖型归约到几何设置。
摘要：
近似Steiner树的最佳算法性能比为ln(4)+ε≈1.386，而不可近似性结果停留在96/95≈1.0105。本文向前迈进以弥合差距，表明除非P = NP，否则不存在多项式时间算法以优于19/18≈1.0555的常数比近似Steiner树。还通过证明找到优于17/16 = 1.0625的常数近似比是UG难的，将该问题与唯一博弈猜想相关联。在准二分图的特殊情况下，证明了除非P = NP，不可近似性因子为25/24≈1.0416，改进了先前的128/127≈1.0078。所有情况的归约都有相同思想并适当修改。主要技术贡献是将使用长码的集合覆盖型归约适配到所考虑问题的几何设置中。
英文摘要：
The best algorithm for approximating Steiner tree has performance ratio $\ln(4)+ε\approx 1.386$ [J. Byrka et al., \textit{Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC)}, 2010, pp. 583-592], whereas the inapproximability result stays at the factor $\frac{96}{95} \approx 1.0105$ [M. Chlebík and J. Chlebíková, \textit{Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT)}, 2002, pp. 170-179]. In this article, we take a step forward to bridge this gap and show that there is no polynomial time algorithm approximating Steiner tree with constant ratio better than $\frac{19}{18} \approx 1.0555$ unless \textsf{P = NP}. We also relate the problem to the Unique Games Conjecture by showing that it is \textsf{UG}-hard to find a constant approximation ratio better than $\frac{17}{16} = 1.0625$. In the special case of quasi-bipartite graphs, we prove an inapproximability factor of $\frac{25}{24} \approx 1.0416$ unless \textsf{P = NP}, which improves upon the previous bound of $\frac{128}{127} \approx 1.0078$. The reductions that we present for all the cases are of the same spirit with appropriate modifications. Our main technical contribution is an adaptation of a Set-Cover type reduction in which the Long Code is used to the geometric setting of the problems we consider.
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