Algorithms for bipartite matching problems with connections to sparsification and streaming [electronic resource]
Keywords: Computational and Mathematical Engineering.

The problem of finding maximum matchings in bipartite graphs is a classical problem in combinatorial optimization with a long algorithmic history. Graph sparsification is a more recent paradigm of replacing a graph with a smaller subgraph that preserves some useful properties of the original graph, perhaps approximately. Traditionally, sparsification has been used for obtaining faster algorithms for cut-based optimization problems. The contributions of this thesis are centered around new algorithms for bipartite matching problems, in which, surprisingly, graph sparsification plays a major role, and efficient algorithms for constructing sparsifiers in modern data models. In the first part of the thesis we develop sublinear time algorithms for finding perfect matchings in regular bipartite graphs. These graphs have been studied extensively in the context of expander constructions, and have several applications in combinatorial optimization. The problem of finding perfect matchings in regular bipartite graphs has seen almost 100 years of algorithmic history, with the first algorithm dating back to K\"onig in 1916 and an algorithm with runtime linear in the number of edges in the graph discovered in 2000. In this thesis we show that, even though traditionally the use of sparsification has been restricted to cut-based problems, in fact sparsification yields extremely efficient {\em sublinear time} algorithms for finding perfect matchings in regular bipartite graphs when the graph is given in adjacency array representation. Thus, our algorithms recover a perfect matching (with high probability) without looking the whole input. We present two approaches, one based on independent sampling and another on random walks, obtaining an algorithm that recovers a perfect matching in $O(n\log n)$ time, within $O(\log n)$ of output complexity, essentially closing the problem. In the second part of the thesis we study the streaming complexity of maximum bipartite matching. This problem is relevant to modern data models, where the algorithm is constrained in space and is only allowed few passes over the input. We are interested in determining the best tradeoff between the space usage and the quality of the solution obtained. We first study the problem in the single pass setting. A central object of our study is a new notion of sparsification relevant to matching problems: we define the notion of an $\e$-matching cover of a bipartite graph as a subgraph that approximately preserves sizes of matchings between every two subsets of vertices, which can be viewed as a 'sparsifier' for matching problems. We give an efficient construction of a sparse subgraph that we call a 'matching skeleton', which we show is a linear-size matching cover for a certain range of parameters (in fact, for $\e> 1/2$). We then show that our 'sparsifier' can be applied repeatedly while maintaining a non-trivial approximation ratio in the streaming model with vertex arrivals, obtaining the first



$ passes, improving upon the previously best known approximation. In the third part of the thesis we consider the concept of {\em spectral sparsification} introduced by Spielman and Teng. Here, we uncover a connection between spectral sparsification and spanners, i.e. subgraphs that approximately preserve shortest path distances. This connection allows us to obtain a quasilinear time algorithm for constructing spectral sparsifiers using approximate distance oracles and entirely bypassing linear system solvers, which was previously the only known way of constructing spectral sparsifiers in quasilinear time. Finally, in the last part of the thesis we design an efficient implementation of cut-preserving sparsification in a streaming setting with edge deletions using only one pass over the data.