多重覆盖问题的近似算法
Approximation 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证明，对于任何固定的$Δ$和$b:=\min_{v\in V}b_{v}$，\sbmultcov的问题在小于$δ:=Δ-b+1$的比率内是不可近似的，除非$\mathcal{P} =\mathcal{NP}$。因此，这是一个挑战，探索哪些类超图猜想不成立。我们提出了一个多项式时间算法的集合多重覆盖问题，它结合了一个确定性的阈值算法与条件随机舍入步骤。我们的算法产生的近似比$ \max\left\{ \frac{148}{149}δ, \left(1- \frac{ (b-1)e^{\fracδ{4}}}{94\ell} \right)δ\right\}$。我们的结果不仅比Srivastav等人提出的近似比有所改进（Mammica 2016），而且由于我们对参数$\ell$没有限制，因此它更一般。此外，我们提出了一个进一步的多项式时间算法的逼近比$\frac{5}{6}δ$的超图与$\ell\leq (1+ε)\bar{\ell}$任何固定的$ε\in [0,\frac{1}{2}]$，其中$\bar{\ell}$是平均边的大小。该算法的分析依赖于Ray-Chaudhuri（1960）的匹配/覆盖对偶，我们将其转换为近似形式。第二个结果证明了peleg等人关于超图的一个大子类的猜想是错误的。
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 for any fixed $Δ$ and $b:=\min_{v\in V}b_{v}$, the problem of \sbmultcov is not approximable within a ratio less than $δ:=Δ-b+1$, unless $\mathcal{P} =\mathcal{NP}$. Hence it's a challenge to explore for which classes of hypergraph the conjecture doesn't hold. We present a polynomial time algorithm for the Set Multicover problem which combines a deterministic threshold algorithm with conditioned randomized rounding steps. Our algorithm yields an approximation ratio of $ \max\left\{ \frac{148}{149}δ, \left(1- \frac{ (b-1)e^{\fracδ{4}}}{94\ell} \right)δ\right\}$. Our result not only improves over the approximation ratio presented by Srivastav et al (Algorithmica 2016) but it's more general since we set no restriction on the parameter $\ell$. Moreover we present a further polynomial time algorithm with an approximation ratio of $\frac{5}{6}δ$ for hypergraphs with $\ell\leq (1+ε)\bar{\ell}$ for any fixed $ε\in [0,\frac{1}{2}]$, where $\bar{\ell}$ is the average edge size. The analysis of this algorithm relies on matching/covering duality due to Ray-Chaudhuri (1960), which we convert into an approximative form. The second performance disprove the conjecture of peleg et al for a large subclass of hypergraphs.
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