加权$k$ -集包装的纯局部搜索极限超越
Passing the Limits of Pure Local Search for Weighted $k$-Set Packing
摘要：
我们研究加权$k$ -Set Packing问题：给定集合$S$，每个集合的基数至多为$k$，加上正的权重函数$w:\mathcal{S}\rightarrow\mathbb{Q}_{>0}$，任务是计算最大总权重的不相交子集合$A\subseteq \mathcal{S}$。对于$k\leq 2$，加权$k$ -集合装箱问题可以在多项式时间内求解，但对于$k\geq 3$，它变成了$NP$ -hard。最近，Neuwohner展示了如何用$\lim_{k\rightarrow\infty}ε_k=0$获得$\frac{k+ε_k}{2}$的近似保证。她进一步表明，她的结果是渐近最好的可能，因为没有算法考虑局部改进的数值界大小相对于一些固定的权力的权重函数可以产生一个近似保证比$\frac{k}{2}$。本文最后给出了如何用$Ω(k)$克服加权$k$ -Set Packing问题的$\frac{k}{2}$的阈值。我们实现这一目标相结合的本地搜索与应用程序的黑箱算法的未加权$k$ -集包装问题仔细选择的子实例。在这样做时，我们设法链接的近似比一般的权重，以实现在未加权的情况下，我们得到的保证，最多$\frac{k+1}{2}-2\cdot 10^{-4}$的所有$k\geq 4$。
英文摘要：
We study the weighted $k$-Set Packing problem: Given a collection $S$ of sets, each of cardinality at most $k$, together with a positive weight function $w:\mathcal{S}\rightarrow\mathbb{Q}_{>0}$, the task is to compute a disjoint sub-collection $A\subseteq \mathcal{S}$ of maximum total weight. For $k\leq 2$, the weighted $k$-Set Packing problem can be solved in polynomial time, but for $k\geq 3$, it becomes $NP$-hard. Recently, Neuwohner has shown how to obtain approximation guarantees of $\frac{k+ε_k}{2}$ with $\lim_{k\rightarrow\infty}ε_k=0$. She further showed her result to be asymptotically best possible in that no algorithm considering local improvements of logarithmically bounded size with respect to some fixed power of the weight function can yield an approximation guarantee better than $\frac{k}{2}$. In this paper, we finally show how to beat the threshold of $\frac{k}{2}$ for the weighted $k$-Set Packing problem by $Ω(k)$. We achieve this by combining local search with the application of a black box algorithm for the unweighted $k$-Set Packing problem to carefully chosen sub-instances. In doing so, we manage to link the approximation ratio for general weights to the one achievable in the unweighted case and we obtain guarantees of at most $\frac{k+1}{2}-2\cdot 10^{-4}$ for all $k\geq 4$.
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