在线二分匹配中的建议权和随机权
On the Power of Advice and Randomization for Online Bipartite Matching
摘要：
虽然随机化在线算法可以访问均匀随机位的序列，但是具有建议的确定性在线算法可以访问建议位的序列，即，在处理请求序列之前由全能的oracle设置的位。建议比特至少和随机比特一样有用，但它们有多大帮助呢？在这项工作中，我们调查的权力的建议位和随机位在线最大二分匹配（MBM）。众所周知的Karp-Vazirani-Vazirani算法是用于\textsc{MBM}的最优随机化$(1-\frac{1}{e})$竞争算法，其需要访问$Θ(n \log n)$均匀随机比特。我们表明，$Ω(\log(\frac{1}ε) n)$咨询位是必要的和$O(\frac{1}{ε^5} n)$足够，以获得一个$(1-ε)$竞争确定性的建议算法。此外，对于一个大的自然类的确定性建议算法，我们证明了$Ω(\log \log \log n)$咨询位，以提高$\frac{1}{2}$ -竞争力的最佳确定性在线算法，而它是已知的，$O(\log n)$位是足够的。最后，我们给出了一个随机的在线算法，使用$c n$随机位，整数$c \ge 1$，和一个竞争比，接近$1-\frac{1}{e}$非常迅速的$c$的增加。例如，如果是$c = 10$，则$1-\frac{1}{e}$和所获得的竞争比之间的差小于$0.0002$。
英文摘要：
While randomized online algorithms have access to a sequence of uniform random bits, deterministic online algorithms with advice have access to a sequence of advice bits, i.e., bits that are set by an all powerful oracle prior to the processing of the request sequence. Advice bits are at least as helpful as random bits, but how helpful are they? In this work, we investigate the power of advice bits and random bits for online maximum bipartite matching (MBM). The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized $(1-\frac{1}{e})$-competitive algorithm for \textsc{MBM} that requires access to $Θ(n \log n)$ uniform random bits. We show that $Ω(\log(\frac{1}ε) n)$ advice bits are necessary and $O(\frac{1}{ε^5} n)$ sufficient in order to obtain a $(1-ε)$-competitive deterministic advice algorithm. Furthermore, for a large natural class of deterministic advice algorithms, we prove that $Ω(\log \log \log n)$ advice bits are required in order to improve on the $\frac{1}{2}$-competitiveness of the best deterministic online algorithm, while it is known that $O(\log n)$ bits are sufficient. Last, we give a randomized online algorithm that uses $c n$ random bits, for integers $c \ge 1$, and a competitive ratio that approaches $1-\frac{1}{e}$ very quickly as $c$ is increasing. For example if $c = 10$, then the difference between $1-\frac{1}{e}$ and the achieved competitive ratio is less than $0.0002$.
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。
