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{\large A Sparsification Based Algorithm for Maximum-Cardinality Bipartite
Matching in Planar Graphs}

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Mudabir Kabir Asathulla

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(ABSTRACT)

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$\textit{Matching}$ is one of the most fundamental algorithmic graph problems. Many variants of matching problems have been studied on different classes of graphs, the one of specific interest to us being the $\textit{Maximum Cardinality}$ $\textit{Bipartite Matching in}$ $\textit{Planar Graphs}$. In this work, we present a novel sparsification based approach for computing maximum/perfect bipartite matching in planar graphs. The overall complexity of our algorithm is $\sixfivefast$ where $n$ is the number of vertices in the graph,  bettering the $O(n^{3/2})$ time achieved independently by Hopcroft-Karp algorithm and by Lipton $\&$ Tarjan \textrevapostrophe divide and conquer' approach using planar separators. Our algorithm combines the best of both these standard algorithms along with our sparsification technique and rich planar graph properties to achieve the speed up. Our algorithm is not the fastest, with the existence of $O(n\log^{3}{n})$ algorithm based on max-flow reduction.
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{\large A Sparsification Based Algorithm for Maximum-Cardinality Bipartite
Matching in Planar Graphs}

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Mudabir Kabir Asathulla

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(GENERAL AUDIENCE ABSTRACT)

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A matching in a graph can be defined as a subset of edges without common vertices. A matching algorithm finds a maximum set of such vertex-disjoint edges. Many real life resource allocation problems can be solved efficiently by modelling them as a matching problem. While many variants of matching problems have been studied on different classes of graphs, the simplest and the most popular among them is the $\textit{Maximum Cardinality Bipartite Matching}$ problem. Bipartite matching arises in varied applications like matching applicants to job openings, matching ads to user queries, matching threads to tasks in OS scheduler, matching  protein sequences based on their structures and so on. 

In this work, we present an efficient algorithm for computing maximum cardinality bipartite matching in planar graphs. Planar graphs are sparse graphs and have interesting structural properties which allow us to design faster algorithms in planar setting for problems that are otherwise considered hard in arbitrary graphs. We use a new sparsification based approach where we maintain a compact and accurate representation of the original graph with a lesser number of vertices.  Our algorithm combines the features of the best known bipartite matching algorithm for an arbitrary graph with the novel sparsification approach to achieve the speedup. 
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{\large \textit{To my beloved family,}}\\
{\large \textit{Mom, Dad and Brother,}}\\
{\large \textit{Thanks for endless love, support and sacrifices.}}
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