捷径与传递闭包扳手的近似
Shortcuts and Transitive-Closure Spanners Approximation
AI总结：
本文研究捷径与传递闭包扳手的近似算法，在投影博弈猜想下证明不存在多项式时间的$(n^\epsilon,n^\epsilon)$-近似算法，并给出满足$3\gamma_D+2\gamma_S>1$的$(n^{\gamma_D},n^{\gamma_S})$-近似上界。
摘要：
我们研究两个密切相关问题的多项式时间近似算法，即计算捷径和传递闭包扳手（TC扳手）。对于有向无权图$G=(V, E)$和整数$d$，若图$H:=(V, E')$满足(i)与$G$具有相同的传递闭包，且(ii)直径至多为$d$，则边集$E'\subseteq V\times V$称为$G$的$d$-TC扳手。若$E\cup E''$是$G$的$d$-TC扳手，则集合$E''\subseteq V\times V$是$G$的$d$-捷径。我们关注以下$(\alpha_D, \alpha_S)$-近似算法：给定有向图$G$和整数$d$与$s$，使得$G$存在大小为$s$的$d$-捷径（相应地，$d$-TC扳手），找到具有$s\alpha_S$条边的$(d\alpha_D)$-捷径（相应地，$(d\alpha_D)$-TC扳手），并尽可能使$\alpha_S$和$\alpha_D$小。作为主要结果，我们证明在投影博弈猜想（PGC）下，存在一个小常数$\epsilon>0$，使得不存在多项式时间的$(n^\epsilon,n^\epsilon)$-近似算法来寻找大小为$s$的$d$-捷径以及$d$-TC扳手。此前，仅对常数$d$且${\alpha_D}=1$的$d$-TC扳手已知超常数下界[Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff 2009]。对于超常数$d$的类似下界，此前仅对更一般的有向扳手情况已知[Elkin, Peleg 2000]。在我们的结果之前，对于捷径问题没有已知的近似难度结果。作为附带贡献，我们用形如$(n^{\gamma_D}, n^{\gamma_S})$-近似的上界补充上述结果，该上界在$3\gamma_D + 2\gamma_S > 1$时成立（例如，$(n^{1/5+o(1)}, n^{1/5+o(1)})$-近似）。
英文摘要：
We study polynomial-time approximation algorithms for two closely-related problems, namely computing shortcuts and transitive-closure spanners (TC spanners). For a directed unweighted graph $G=(V, E)$ and an integer $d$, a set of edges $E'\subseteq V\times V$ is called a $d$-TC spanner of $G$ if the graph $H:=(V, E')$ has (i) the same transitive-closure as $G$ and (ii) diameter at most $d.$ The set $E''\subseteq V\times V$ is a $d$-shortcut of $G$ if $E\cup E''$ is a $d$-TC spanner of $G$. Our focus is on the following $(α_D, α_S)$-approximation algorithm: given a directed graph $G$ and integers $d$ and $s$ such that $G$ admits a $d$-shortcut (respectively $d$-TC spanner) of size $s$, find a $(dα_D)$-shortcut (resp. $(dα_D)$-TC spanner) with $sα_S$ edges, for as small $α_S$ and $α_D$ as possible. As our main result, we show that, under the Projection Game Conjecture (PGC), there exists a small constant $ε>0$, such that no polynomial-time $(n^ε,n^ε)$-approximation algorithm exists for finding $d$-shortcuts as well as $d$-TC spanners of size $s$. Previously, super-constant lower bounds were known only for $d$-TC spanners with constant $d$ and ${α_D}=1$ [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff 2009]. Similar lower bounds for super-constant $d$ were previously known only for a more general case of directed spanners [Elkin, Peleg 2000]. No hardness of approximation result was known for shortcuts prior to our result. As a side contribution, we complement the above with an upper bound of the form $(n^{γ_D}, n^{γ_S})$-approximation which holds for $3γ_D + 2γ_S > 1$ (e.g., $(n^{1/5+o(1)}, n^{1/5+o(1)})$-approximation).
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