覆盖不相容集合族的近似算法与参数化算法
Approximation and parameterized algorithms for covering disjointness-compliable set families
AI总结：
研究了覆盖不相容可补集合族的近似算法，证明其近似比为O(α log τ)，并给出了多根覆盖Steiner问题的近似算法。
摘要：
一个集合族F是不相容可补的，如果A'⊆A∈F则意味着A'∈F或A\A'∈F；如果F也是对称的，则F是proper的。Goemans和Williamson的经典结果表明，用最小成本边集覆盖proper集合族的问题具有2的近似比，通过经典的对偶算法。然而，有许多著名的算法问题其集合族F是不相容可补但非对称的——其中包括k-最小生成树（k-MST）、一般点对点连接（G-P2P）、组Steiner、覆盖Steiner、这些问题的多根版本等。我们将证明任何此类问题具有O(α log τ)的近似比，其中τ是集合族F中表示问题的最小包含集合的数量，α是当τ=1时的最佳近似比。这立即推导出几个结果，其中包括以下两个。 (i) G-P2P问题的首个确定性多项式时间O(log n)近似算法。这里τ=1的情况是k-MST问题。(ii) 覆盖Steiner问题的多根版本的近似比为O(log⁴n)，其中每个根有其自己的组集。这里τ=1的情况是覆盖Steiner问题。我们还讨论了当参数化为τ时覆盖不相容可补族F的参数化复杂性。我们将证明如果F是proper的，则该问题是固定参数可 tractable 的，并可在时间O*(3^τ)内解决。对于非对称情况，我们将证明该问题在时间O*(3^τ)内具有近似比在α和α+1之间，这基本上是最佳的。
英文摘要：
A set-family ${\cal F}$ is disjointness-compliable if $A' \subseteq A \in {\cal F}$ implies $A' \in {\cal F}$ or $A \setminus A' \in {\cal F}$; if ${\cal F}$ is also symmetric then ${\cal F}$ is proper. A classic result of Goemans and Williamson [SODA 92:307-316] states that the problem of covering a proper set-family by a min-cost edge set admits approximation ratio $2$, by a classic primal-dual algorithm. However, there are several famous algorithmic problems whose set-family ${\cal F}$ is disjointness-compliable but not symmetric -- among them $k$-Minimum Spanning Tree ($k$-MST), Generalized Point-to-Point Connection (G-P2P), Group Steiner, Covering Steiner, multiroot versions of these problems, and others. We will show that any such problem admits approximation ratio $O(α\log τ)$, where $τ$ is the number of inclusion-minimal sets in the family ${\cal F}$ that models the problem and $α$ is the best known approximation ratio for the case when $τ=1$. This immediately implies several results, among them the following two. (i) The first deterministic polynomial time $O(\log n)$-approximation algorithm for the G-P2P problem. Here the $τ=1$ case is the $k$-MST problem. (ii) Approximation ratio $O(\log^4 n)$ for the multiroot version of the Covering Steiner problem, where each root has its own set of groups. Here the $τ=1$ case is the Covering Steiner problem. We also discuss the parameterized complexity of covering a disjointness-compliable family ${\cal F}$, when parametrized by $τ$. We will show that if ${\cal F}$ is proper then the problem is fixed parameter tractable and can be solved in time $O^*(3^τ)$. For the non-symmetric case we will show that the problem admits approximation ratio between $α$ and $α+1$ in time $O^*(3^τ)$, which is essentially the best possible.
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