In the weighted bipartite matching problem, the goal is to find a maximum-weight matching in a bipartite graph with nonnegative edge weights. We consider its online version where the first vertex set is known beforehand, but vertices of the second set appear one after another. Vertices of the first set are interpreted as items, and those of the second set as bidders. On arrival, each bidder vertex reveals the weights of all adjacent edges and the algorithm has to decide which of those to add to the matching. We introduce an optimal, $e$-competitive truthful mechanism under the assumption that bidders arrive in random order (secretary model).
It has been shown that the upper and lower bound of e for the original secretary problem extends to various other problems even with rich combinatorial structure, one of them being weighted bipartite matching. But truthful mechanisms so far fall short of reasonable competitive ratios once respective algorithms deviate from the original, simple threshold form. The best known mechanism for weighted bipartite matching by Krysta and Vöcking (ICALP 2012) offers only a ratio logarithmic in the number of online vertices. We close this gap, showing that truthfulness does not impose any additional bounds. The proof technique is new in this surrounding, and based on the observation of an independency inherent to the mechanism. The insights provided hereby are interesting in their own right and appear to offer promising tools for other problems, with or without truthfulness.
Truthful Mechanisms for Delivery with Mobile Agents
我们研究了博弈论的任务，选择移动代理提供多个项目在网络上。一个实例由$m$消息（物理对象）和$k$移动异构代理给出，这些消息必须在加权无向图中的指定源-目标对之间传输，每个移动异构代理都能够一次传输一个消息。根据[Bärtschi et al. 2016]的最新模型，每个代理$i$消耗的能量与它在图中行进的距离成比例，其中不同的能量消耗率由权重因子$w_i$给出。我们感兴趣的是优化或近似所有选定的代理的总能耗。与以往的研究不同，我们假设权重是私有值只知道各自的代理。我们提出了三种不同的机制，选择，路由和支付的代理人在一个真实的方式，保证自愿参与的代理人，而近似的最佳能源消耗的一个常数的因素。为此，我们分析了以前的结构结果和[Bärtschi et al. 2017]给出的近似算法。最后，我们表明，在某些情况下，在一个单一的包的情况下，付款的总和可以有界的最佳。
Prophet inequality for bipartite matching: merits of being simple and non adaptive
考虑二部图中匹配的贝叶斯在线选择问题，即，带边到达的在线加权匹配问题，其中在线算法知道权值的分布，对应于[Kleinberg and Wienberg STOC 12]模型中两个拟阵的交集。我们考虑一类简单的非自适应顶点添加剂的政策，分配静态价格的所有顶点在图中，并接受每个边，只有当它的重量超过边缘的端点的价格之和。我们表明存在一个顶点添加剂的政策，预期的回报至少有三分之一的先知的回报和目前的梯度下降型算法，迅速收敛到所需的顶点价格的向量。这在两个方面改进了[Kleinberg and Wienberg STOC 12]中关于两个拟阵交集的自适应在线策略：我们的策略是非自适应的，并且具有更好的$3$近似保证，而不是以前的$5.82$对先知的保证。我们给出了一个互补的下限$2.25$的任何在线算法中的二分匹配设置。
