The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to design online algorithms with small competitive ratios.
For a variety of combinatorial problems, such as bipartite matching, general packing LPs, and independent set with bounded local independence number, we design new algorithms that obtain constant competitive ratios.
For the matroid secretary problem, we observe that many existing algorithms for special matroid structures maintain their competitive ratios even in the ordinal model. In these cases, the restriction to ordinal information does not represent any additional obstacle. Moreover, we show that ordinal variants of the submodular matroid secretary problems can be solved using algorithms for the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we provide a lower bound of $Ω(\sqrt{n}/(\log n))$ for algorithms that are oblivious to the matroid structure, where $n$ is the total number of elements. This contrasts an upper bound of $O(\log n)$ in the cardinal model, and it shows that the technique of thresholding is not sufficient for good algorithms in the ordinal model.
A 3.1462-Competitive Algorithm for Matroid Secretary
University of Nebraska–Lincoln（内布拉斯加大学林肯分校）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；Beijing Normal University–Zhuhai（北京师范大学珠海校区）；Beijing Normal–Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）
MTA-ELTE Matroid Optimization Research Group and HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University（MTA-ELTE拟阵优化研究组和HUN-REN-ELTE埃格瓦里研究组，运筹学系，罗兰大学）；Center for Mathematical Modeling, Universidad de Chile（数学建模中心，智利大学）；Department of Mathematical Engineering and Center for Mathematical Modeling, Universidad de Chile（数学工程系和数学建模中心，智利大学）；Institute for Mathematical and Computational Engineering, and Department of Industrial and Systems Engineering, PUC Chile（数学与计算工程研究所和工业与系统工程系，智利天主教 Pontificia Universidad Católica）
Improved Online Algorithms for Knapsack and GAP in the Random Order Model
背包问题是组合优化中的经典问题之一：给定一组物品，每个物品由其大小和利润指定，目标是找到一个最大利润的包装到一个有界容量的背包中。在在线设置中，项目是一个接一个地显示和决定，如果当前项目是包装或永远丢弃，必须立即和可追溯地完成后到达。研究了随机序模型中的在线变量，其中输入序列是项目集的均匀随机排列。我们为这个问题开发了一个随机（1/6.65）竞争算法，优于目前竞争比为1/8.06的最佳算法[Kesselheim et al. SIAM J. Comp. 47（5）]。我们的算法是基于两个新的见解：我们引入了一种新的算法方法，采用两个给定的算法，优化的限制项目类，依次输入序列。此外，我们还研究和探索了背包问题与2-秘书问题之间的关系。广义指派问题（GAP）除了背包问题外，还包括与调度和匹配相关的几个重要问题。我们表明，在相同的在线设置，应用建议的顺序方法产生（1/6.99）的竞争随机算法的GAP。同样，我们提出的算法优于竞争比1/8.06的当前最佳结果[Kesselheim et al. SIAM J. Comp. 47（5）]。
