多重覆盖问题的重复随机算法
Repeated randomized algorithm for the Multicovering Problem
摘要：
设$\mathcal{H}=(V,\mathcal{E})$是具有最大边长$\ell$和最大度$Δ$的超图.对于给定的数$b_v\in \mathbb{N}_{\geq 2}$，$v\in V$，$\mathcal{H}$中的集合多重覆盖是一组边$C \subseteq \mathcal{E}$，使得$V$中的每个顶点$v$至少属于$C$中的$b_v$条边。集合多重覆盖是寻找最小基数集合多重覆盖的问题。Peleg、Schechtman和Wool指出，除非$\cal{P} =\cal{NP}$，对于任何固定的$Δ$和$b:=\min_{v\in V}b_{v}$，没有多项式时间近似算法的集合多重覆盖问题的近似比小于$δ:=Δ-b+1$。因此，这是一个挑战，知道是否集多重覆盖的问题是不是近似的$βδ$与常数$β<1$的比率。本文提出了一种结合初始确定性阈值步长的集合多重覆盖问题的重复随机算法。通过反复试验提高成功率，我们的算法产生了$ \max\left\{ \frac{15}{16}δ, \left(1- \frac{(b-1)\exp\left(\frac{ 3δ+1}{8}\right)}{72 \ell} \right)δ\right\}$的近似比。关键的事实是，我们的结果不仅比Srivastav等人（Mammica 2016）提出的任何$δ\geq 13$的近似比有所改进，而且由于我们对参数$\ell$没有限制，因此它更一般。进一步证明了在$(δ-1-ε)$的一个因子内逼近$Δ$ -正则超图上的集合多重覆盖问题是NP-困难的。此外，我们证明了集合多重覆盖问题的完整性间隙至少是$\frac{\ln_{2}(n+1)}{2b}$，对于常数$b$是$Ω(\ln n )$。
英文摘要：
Let $\mathcal{H}=(V,\mathcal{E})$ be a hypergraph with maximum edge size $\ell$ and maximum degree $Δ$. For given numbers $b_v\in \mathbb{N}_{\geq 2}$, $v\in V$, a set multicover in $\mathcal{H}$ is a set of edges $C \subseteq \mathcal{E}$ such that every vertex $v$ in $V$ belongs to at least $b_v$ edges in $C$. Set multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that unless $\cal{P} =\cal{NP}$, for any fixed $Δ$ and $b:=\min_{v\in V}b_{v}$, no polynomial-time approximation algorithm for the Set multicover problem has an approximation ratio less than $δ:=Δ-b+1$. Hence, it's a challenge to know whether the problem of set multicover is not approximable within a ratio of $βδ$ with a constant $β<1$. This paper proposes a repeated randomized algorithm for the Set multicover problem combined with an initial deterministic threshold step. Boosting success by repeated trials, our algorithm yields an approximation ratio of $ \max\left\{ \frac{15}{16}δ, \left(1- \frac{(b-1)\exp\left(\frac{ 3δ+1}{8}\right)}{72 \ell} \right)δ\right\}$. The crucial fact is not only that our result improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) for any $δ\geq 13$, but it's more general since we set no restriction on the parameter $\ell$. Furthermore, we prove that it is NP-hard to approximate the Set multicover problem on $Δ$-regular hypergraphs within a factor of $(δ-1-ε)$. Moreover we show that the integrality gap for the Set multicover problem is at least $\frac{\ln_{2}(n+1)}{2b}$, which for constant $b$ is $Ω(\ln n )$.
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