## Fractional Matching via Balls-and-Bins

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Citations: | 1 - 0 self |

### BibTeX

@MISC{Motwani_fractionalmatching,

author = {Rajeev Motwani and Rina Panigrahy and Ying Xu},

title = {Fractional Matching via Balls-and-Bins},

year = {}

}

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### Abstract

In this paper we relate the problem of finding structures related to perfect matchings in bipartite graphs to a stochastic process similar to throwing balls into bins. Given a bipartite graph with n nodes on each side, we view each node on the left as having balls that it can throw into nodes on the right (bins) to which it is adjacent. If each node on the left throws exactly one ball and each bin on the right gets exactly one ball, then the edges represented by the ball-placement form a perfect matching. Further, if each thrower is allowed to throw a large but equal number of balls, and each bin on the right receives an equal number of balls, then the set of ball-placements corresponds to a perfect fractional matching – a weighted subgraph on all nodes with nonnegative weights on edges so that the total weight incident at each node is 1. We show that several simple algorithms based on throwing balls into bins deliver a near-perfect fractional matching. For example, we show that by iteratively picking a random node on the left and throwing a ball into its least-loaded neighbor, the distribution of balls obtained is no worse than randomly throwing kn balls into n bins. Another algorithm is based on the d-choice load-balancing of balls and bins. By picking a constant number of nodes on the left and appropriately inserting a ball into the least-loaded of their neighbors, we achieve a smoother load distribution on both sides – maximum load is at most log log n / log d + O(1). When each vertex on the left throws k balls, we obtain an algorithm that achieves a load within k ± 1 on the right vertices. 1

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Citation Context ...matchings in unweighted bipartite graphs using algorithms based on throwing balls into bins. While the problem of finding matchings in graphs is wellstudied [19], as is the balls-and-bins formulation =-=[25]-=-, this paper explores a novel connection between the two problems. A perfect matching is a subgraph on all nodes where every node has degree exactly 1. The problem of finding perfect matchings in bipa... |

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Citation Context ...o each node, unlike other algorithms based on augmenting paths. There are also extensive references on approximation algorithms for multi-commodity flow and generalized flow problems, see for example =-=[12, 13, 14, 28]-=-. While some of those algorithms when applied to the matching problem may result in an algorithm similar to our first algorithm in spirit, we propose a much simpler framework which admits simpler and ... |

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Citation Context ... ports over time and need to be continuously matched to output ports. The k-matching problem in general (weighted) graphs arises in several important applications including the Chinese postman problem=-=[10]-=-, capacitated vehicle routing[22], and quadrilateral mesh refinement in computer-aided design [24]. The stochastic process of throwing balls into random bins is also a well-studied problem (see, for e... |

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Citation Context ...eft nodes) on a search engine such as Google are matched to advertisements (right nodes) based on a set of ad-words; k-matching has been used as a crucial component in several auction design problems =-=[29, 33]-=-; fractional matchings were also used by Azar and Litichevskey [3] to model switch scheduling problem where packets arrive at input ports over time and need to be continuously matched to output ports.... |

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Citation Context ...he problem of finding perfect (fractional) matchings in unweighted bipartite graphs using algorithms based on throwing balls into bins. While the problem of finding matchings in graphs is wellstudied =-=[19]-=-, as is the balls-and-bins formulation [25], this paper explores a novel connection between the two problems. A perfect matching is a subgraph on all nodes where every node has degree exactly 1. The p... |

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Citation Context ...ontinuously matched to output ports. The k-matching problem in general (weighted) graphs arises in several important applications including the Chinese postman problem[10], capacitated vehicle routing=-=[22]-=-, and quadrilateral mesh refinement in computer-aided design [24]. The stochastic process of throwing balls into random bins is also a well-studied problem (see, for example, [23, 2, 4, 30, 34]). The ... |

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