闭上眼睛匹配
Matching with our Eyes Closed
摘要：
受肾脏交换中的应用启发，我们研究了以下查询提交问题：给定一个非二部图G的顶点集。这个图中的边的集合是事先不知道的。我们可以查询任何一对顶点，以确定它们是否相邻。如果查询的边存在，我们将致力于匹配两个端点。我们的目标是最大化匹配的大小。这种对算法可用信息量的限制了我们实现短视的、贪婪的算法。一个简单的确定性贪心算法达到了1/2的因子，这对于确定性算法来说是紧的。在这个方向上的一个重要的开放问题是给一个随机的贪心算法，有一个显着更好的近似因子。这个问题最早是在20年前由Dyer和Frieze提出的，他们证明了随机均匀地拾取边的自然随机策略没有帮助，并且近似因子为1/2 + o（1）。他们把它作为一个开放的问题，设计一个更好的随机贪心算法。在随后的工作中，Aronson，Dyer，Frieze和Suen [2]给出了一个不同的随机贪心算法，并表明它达到了因子0.5 + λ，其中λ为0.0000025。在本文中，我们提出并分析了一个新的随机贪心算法，以寻找一个大的匹配在一般图，并使用它来解决上述的查询提交问题。我们表明，我们的算法达到了至少0.56，一个显着的改善0.50000025的一个因素。我们还表明，没有随机算法可以有一个近似因子优于0.7916的查询提交问题。对于另一个大的和有趣的类的随机算法，我们称之为顶点迭代算法，我们表明，没有顶点迭代算法可以有一个近似因子优于0.75。
英文摘要：
Motivated by an application in kidney exchange, we study the following query-commit problem: we are given the set of vertices of a non-bipartite graph G. The set of edges in this graph are not known ahead of time. We can query any pair of vertices to determine if they are adjacent. If the queried edge exists, we are committed to match the two endpoints. Our objective is to maximize the size of the matching. This restriction in the amount of information available to the algorithm constraints us to implement myopic, greedy-like algorithms. A simple deterministic greedy algorithm achieves a factor 1/2 which is tight for deterministic algorithms. An important open question in this direction is to give a randomized greedy algorithm that has a significantly better approximation factor. This question was first asked almost 20 years ago by Dyer and Frieze [9] where they showed that a natural randomized strategy of picking edges uniformly at random doesn't help and has an approximation factor of 1/2 + o(1). They left it as an open question to devise a better randomized greedy algorithm. In subsequent work, Aronson, Dyer, Frieze, and Suen [2] gave a different randomized greedy algorithm and showed that it attains a factor 0.5 + epsilon where epsilon is 0.0000025. In this paper we propose and analyze a new randomized greedy algorithm for finding a large matching in a general graph and use it to solve the query commit problem mentioned above. We show that our algorithm attains a factor of at least 0.56, a significant improvement over 0.50000025. We also show that no randomized algorithm can have an approximation factor better than 0.7916 for the query commit problem. For another large and interesting class of randomized algorithms that we call vertex-iterative algorithms, we show that no vertex-iterative algorithm can have an approximation factor better than 0.75.
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