Online matching with blocked input

Abstract

In this paper, we examine the problem of “blocked online bipartite matching”. This problem is similar to the online matching problem except that the vertices arrive in blocks instead of one at a time. Previously studied problems exist as special cases of this problem; the case where each block contains only a single vertex is the standard online matching problem studied by Karp et al. (1990), and the case where there is only one block (containing all vertices of the graph) is the offline matching problem (see, for example, the work by Aho et al. (1985)). The main result of this paper is that no performance gain (except in low-order terms) is possible by revealing the vertices in blocks, unless the number of blocks remains constant as n (the number of vertices) grows. Specifically, we show that if the number of vertices in a block is k = o( n), then the expected size of the matching produced by any algorithm (on its worst-case input) is at most (1− 1 e )n + o(n) . This is exactly the bound achieved in the original online matching problem, so no improvement is possible when k = o( n). This result follows from a more general upper bound that applies for all k⩽ n; however, the bound does not appear to be tight for some values of k which are a constant fraction of n (in particular, for k = n 3 ). We also give an algorithm that makes use of the blocked structure of the input. On inputs with k = o( n), this algorithm can be shown to perform at least as well as using the algorithm from Karp et al. (1990) and ignoring blocking. Hence, by the upper bound, our algorithm is optimal to low-order terms for k = o( n), and in some cases considerably outperforms the algorithm of Karp et al. (1990). The algorithm also trivially has optimal performance for k = n; furthermore, it appears to have optimal performance for k = n 2 , but a proof of this performance has not been found. Unfortunately, the algorithm does not meet the upper bound for all block sizes, as is shown by a simple example with block size n 3 . We conjecture that the algorithm we present is actually optimal, and that the upper bound is not tight.