arXiv 2007.049372020-07cs.DS

实用简化子模极大化

Practical Budgeted Submodular Maximization

Moran Feldman, Zeev Nutov, Elad Shoham

备注：: 31 pages, 1 figure

摘要：

本文研究了背包约束下的非负单调次模函数的最大化问题，也称为约束次模最大化问题。Sviridenko（2004）指出，通过猜测最优解的3个适当元素，然后执行贪婪算法，可以获得BSM的$α=1-1/e\approx 0.632$的最佳逼近比。然而，需要猜测（通过枚举）3个元素使得Sviridenko的算法不切实际，因为它导致时间复杂度为$O(n^5)$（可以使用Badanidiyuru & Vondrak（2014）的阈值技术稍微改进，但仅大致为$O(n^4)$）。本文的主要结果表明，较少的猜测就足够了。具体地说，通过仅进行2次猜测，我们得到了相同的$α$的最佳逼近比，其时间复杂度大约为$O(n^3)$。此外，通过仅进行一次猜测，我们在大约$O(n^2)$时间内得到了几乎同样好的$0.6174>0.9767α$近似比。在我们的工作之前，已知的唯一能够获得接近BSM的$α$近似比的算法是Sviridenko的算法和Ene & Nguyen（2019）的算法，该算法实现了$(α-ε)$近似。然而，Ene和Nguyen的算法需要${(1/ε)}^{O(1/ε^4)}n\log^2 n$时间，因此，仅在${(1/ε)}^{O(1/ε^4)}$很大时才具有理论意义，即使对于中等的$ε$值。相比之下，我们分析的所有算法都是简单的和可并行的，这使得它们成为实际使用的好候选者。最近，Tang et al.（2020）研究了一个已经有很长研究历史的简单贪婪算法，并证明了其近似比至少为0.405。我们改进了这个结果，并表明该算法的近似比在[0.427，0.462]范围内。

英文摘要：

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $α=1-1/e\approx 0.632$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko impractical as it leads to a time complexity of $O(n^5)$ (which can be slightly improved using the thresholding technique of Badanidiyuru & Vondrak (2014), but only to roughly $O(n^4)$). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses, we get the same optimal approximation ratio of $α$ with an improved time complexity of roughly $O(n^3)$. Furthermore, by making only a single guess, we get an almost as good approximation ratio of $0.6174>0.9767α$ in roughly $O(n^2)$ time. Prior to our work, the only algorithms that were known to obtain an approximation ratio close to $α$ for BSM were the algorithm of Sviridenko and an algorithm of Ene & Nguyen (2019) that achieves $(α-ε)$-approximation. However, the algorithm of Ene & Nguyen requires ${(1/ε)}^{O(1/ε^4)}n\log^2 n$ time, and hence, is of theoretical interest only as ${(1/ε)}^{O(1/ε^4)}$ is huge even for moderate values of $ε$. In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (2020) studied a simple greedy algorithm that already has a long research history, and proved that its approximation ratio is at least 0.405. We improve over this result, and show that the approximation ratio of this algorithm is within the range [0.427, 0.462].

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文