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Let $A$ and $B$ be two point sets in the plane of sizes $r$ and $n$ respectively (assume $r leq n$), and let $k$ be a parameter. A matching between $A$ and $B$ is a family of pairs in $A times B$ so that any point of $A cup B$ appears in at most one pair. Given two positive integers $p$ and $q$, we define the cost of matching $M$ to be $c(M) = sum_{(a, b) in M}|{a-b}|_p^q$ where $|{cdot}|_p$ is the $L_p$-norm. The geometric partial matching problem asks to find the minimum-cost size-$k$ matching between $A$ and $B$. We present efficient algorithms for geometric partial matching problem that work for any powers of $L_p$-norm matching objective: An exact algorithm that runs in $O((n + k^2) {mathop{mathrm{polylog}}} n)$ time, and a $(1 + varepsilon)$-approximation algorithm that runs in $O((n + ksqrt{k}) {mathop{mathrm{polylog}}} n cdot logvarepsilon^{-1})$ time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in $O(min{n^2, rn^{3/2}} {mathop{mathrm{polylog}}} n)$ time.
The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contac
We give a quasipolynomial time algorithm for the graph matching problem (also known as noisy or robust graph isomorphism) on correlated random graphs. Specifically, for every $gamma>0$, we give a $n^{O(log n)}$ time algorithm that given a pair of $ga
Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with a
We consider the problem of finding textit{semi-matching} in bipartite graphs which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted case. For the weighted case,
For over a decade now we have been witnessing the success of {em massive parallel computation} (MPC) frameworks, such as MapReduce, Hadoop, Dryad, or Spark. One of the reasons for their success is the fact that these frameworks are able to accurately