We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e 'barrier' in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k > 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 − 1 k − 1 k 2 + 1 n -in particular Ranking achieves a factor of 1 − o(1) for graphs with ω(1) disjoint perfect matchings.

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Abstract-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 + ǫ where ǫ 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 vertexiterative algorithm can have an approximation factor better than 0.75.

Abstract-We present a robust algorithm for estimating visibility from a given viewpoint for a point set containing concavities, non-uniformly spaced samples, and possibly corrupted with noise. Instead of performing an explicit surface reconstruction for the points set, visibility is computed based on a construction involving convex hull in a dual space, an idea inspired by the work of Katz et al. [26]. We derive theoretical bounds on the behavior of the method in the presence of noise and concavities, and use the derivations to develop a robust visibility estimation algorithm. In addition, computing visibility from a set of adaptively placed viewpoints allows us to generate locally consistent partial reconstructions. Using a graph based approximation algorithm we couple such reconstructions to extract globally consistent reconstructions. We test our method on a variety of 2D and 3D point sets of varying complexity and noise content.