[论文解读] The Submodular Secretary Problem Goes Linear

Moran Feldman, Rico Zenklusen|arXiv (Cornell University)|Jul 30, 2015

Optimization and Search Problems参考文献 22被引用 12

一句话总结

本文提出了从子模性拟阵秘书问题（SMSP）到经典拟阵秘书问题（MSP）的黑箱归约，证明了针对某类拟阵的任何O(1)-竞争性算法，若适用于MSP，则可导出适用于同一类拟阵的SMSP的O(1)-竞争性算法。关键结果是：拟阵秘书问题猜想（MSP的O(1)-竞争性）与SMSP的同一猜想等价，意味着SMSP在本质上并不比MSP更困难。

ABSTRACT

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most questions related to SMSP remained open, including whether SMSP may be substantially more difficult than MSP; and more generally, to what extend MSP and SMSP are related. Our goal is to address these points by presenting general black-box reductions from SMSP to MSP. In particular, we show that any O(1)-competitive algorithm for MSP, even restricted to a particular matroid class, can be transformed in a black-box way to an O(1)-competitive algorithm for SMSP over the same matroid class. This implies that the matroid secretary conjecture is equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a particular matroid class, it suffices to consider MSP over the same matroid class. Using our reductions we obtain many first and improved O(1)-competitive algorithms for SMSP over various matroid classes by leveraging known algorithms for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive algorithm for SMSP, thus, matching the currently best asymptotic algorithm for MSP, and substantially improving on the previously best O(log(rank))-competitive algorithm for SMSP.

研究动机与目标

• 解决子模性拟阵秘书问题（SMSP）是否在本质上比经典拟阵秘书问题（MSP）更困难的问题。
• 建立SMSP到MSP的一般性黑箱归约，实现竞争性算法的转移。
• 研究MSP与SMSP在竞争比保证和算法可行性方面的关系。
• 通过利用MSP中的已知结果，改进SMSP的渐近竞争比。
• 确定拟阵秘书问题猜想是否蕴含SMSP的同一猜想。

提出的方法

• 设计一种黑箱归约，将任何针对MSP的α-竞争性算法转化为同一拟阵类上SMSP的O(α²)-竞争性算法。
• 提出一种随机化算法（算法2），通过精心选择的概率p = (3α)⁻¹来平衡选择与子模函数收益。
• 采用基于子模函数在随机子集上期望值的概率分析，利用子模性与非负性。
• 使用关键引理，将子集并集上子模函数的期望值与各独立期望之和关联起来，利用子模性。
• 通过独立 oracle 模型应用集中与期望界，推导竞争比保证。
• 依赖于任意MSP实例中存在一组相关随机独立集{P_i}，其总期望权重至少为w(OPT)/α。

实验结果

研究问题

• RQ1 子模性拟阵秘书问题（SMSP）是否严格比经典拟阵秘书问题（MSP）更困难？
• RQ2 能否将任意给定拟阵类上MSP的O(1)-竞争性算法转化为同一类SMSP的O(1)-竞争性算法？
• RQ3 拟阵秘书问题猜想（MSP的O(1)-竞争性）是否蕴含SMSP的同一猜想？
• RQ4 SMSP的最佳渐近竞争比是多少？与MSP的最佳已知结果相比如何？
• RQ5 拟阵与子模函数的结构性质在不确定环境下的在线选择中相互作用的程度如何？

主要发现

• 拟阵秘书问题猜想与SMSP的同一猜想等价，意味着SMSP在本质上并不比MSP更困难。
• 任何针对某类拟阵的MSP的O(1)-竞争性算法，均可通过黑箱归约转化为同一类拟阵上SMSP的O(1)-竞争性算法。
• 实现了O(log log(rank))-竞争性的SMSP算法，与MSP的最佳已知渐近竞争比一致。
• 此前SMSP的最佳已知竞争比为O(log(rank))；本文通过归约将其改进为O(log log(rank))。
• 当MSP算法的竞争比为α时，SMSP结果算法的竞争比为O(kα(3α+1))，其中k为MSP解中相关集合的数量。
• 该归约框架首次实现了针对多种拟阵类的SMSP的O(1)-竞争性算法，并实现了更优的性能，其基础是现有MSP算法。