Conference Publications  
Hypergraph $k$Cut in Randomized Polynomial Time In the hypergraph $k$cut problem, the input is a hypergraph, and the goal is to find a smallest subset of hyperedges whose removal ensures that the remaining hypergraph has at least $k$ connected components. This problem is known to be at least as hard as the densest $k$subgraph problem when k is part of the input (ChekuriLi, 2015). We present a randomized polynomial time algorithm to solve the hypergraph $k$cut problem for constant $k$. Our algorithm solves the more general hedge $k$cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge $k$cut problem, the input is a hedgegraph specified by a vertex set and a disjoint set of hedges, where each hedge is a subset of edges defined over the vertices. The goal is to find a smallest subset of hedges whose removal ensures that the number of connected components in the remaining underlying (multi)graph is at least $k$. Our algorithm is based on random contractions akin to Karger's min cut algorithm. Our main technical contribution is a distribution over the hedges (hyperedges) so that random contraction of hedges (hyperedges) chosen from the distribution succeeds in returning an optimum solution with large probability. 

Global and fixedterminal cuts in digraphs
The computational complexity of multicutlike problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.


A Faster Pseudopolynomial Time Algorithm for Subset Sum
Given a multiset $S$ of $n$ positive integers and a target integer $t$, the subset sum problem is to decide if there is a subset of $S$ that sums up to $t$. We present a new divideandconquer algorithm that computes all the realizable subset sums up to an integer $u$ in $\tilde{O}\left(\min\{n\sqrt{u},u^{4/3},\sigma\}\right)$, where $\sigma$ is the sum of all elements in $S$ and $\tilde{O}$ hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in $O(nu)$ time. To the best of our knowledge, the new algorithm is the fastest general algorithm for this problem. We also present a modified algorithm for cyclic groups, which computes all the realizable subset sums within the group in $\tilde{O}\left(\min\{n\sqrt{m},m^{5/4}\}\right)$ time, where m is the order of the group.


Computing minimum cuts in hypergraphs
We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n=V$, $m=E$ and $p=\sum_{e\in E}e$ the best known algorithm to compute a global minimum cut in $H$ runs in time $O(np)$ for the uncapacitated case and in $O(np+n^2\log n)$ time for the capacitated case. We show the following new results.


On ElementConnectivity Preserving Graph Simplification
The notion of elementconnectivity has found several important applications in network design and routing problems. We focus on a reduction step that preserves the elementconnectivity, which when applied repeatedly allows one to reduce the original graph to a simpler one. This preprocessing step is a crucial ingredient in several applications. In this paper we revisit this reduction step and provide a new proof via the use of setpairs. Our main contribution is algorithmic results for several basic problems on elementconnectivity including the problem of achieving the aforementioned graph simplification. We utilize the underlying submodularity properties of elementconnectivity to derive faster algorithms.


Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the Fréchet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in $O(n \log n)$ time, improving an earlier $O(n^3)$time algorithm of Cortese et al.. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary nvertex polygon is weakly simple; our algorithm runs in $O(n^2 \log n)$ time. We also describe algorithms that detect weak simplicity in $O(n \log n)$ time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.
Dedicated with thanks to our colleague Ferran Hurtado (1951–2014). 

Journal Publications  
The shortest kinship description problem
We consider a problem in descriptive kinship systems, namely finding the shortest sequence of terms that describes the kinship between a person and his/her relatives. The problem reduces to finding the minimum weight path in a labeled graph where the label of the path comes from a regular language. The running time of the algorithm is $O(n^3+s)$, where $n$ and $s$ are the input size and the output size of the algorithm, respectively.
To the memories of Jiaqi Zhao(1994–2016). 

Beating the 2approximation factor for global bicut
In the fixedterminal bicut problem, the input is a directed graph with two specified nodes s and t and the goal is to find a smallest subset of edges whose removal ensures that s cannot reach t and t cannot reach s. In the global bicut problem, the input is a directed graph and the goal is to find a smallest subset of edges whose removal ensures that there exist two nodes s and t such that s cannot reach t and t cannot reach s. Fixedterminal bicut and global bicut are natural extensions of $\{s,t\}$min cut and global mincut respectively, from undirected graphs to directed graphs. Fixedterminal bicut is NPhard, admits a simple $2$approximation, and does not admit a $(2−\epsilon)$approximation for any constant $\epsilon>0$ assuming the unique games conjecture. In this work, we show that global bicut admits a $(2−1/448)$approximation, thus improving on the approximability of the global variant in comparison to the fixedterminal variant.


Reconstructing edgedisjoint paths faster
For a simple undirected graph with $n$ vertices and $m$ edges, we consider a data structure that given a query of a pair of vertices $u$, $v$ and an integer $k\geq 1$, it returns $k$ edgedisjoint $uv$paths. The data structure takes $\tilde{O}(n^{3.375})$ time to build, using $O(mn^{1.5}\log n)$ space, and each query takes $O(kn)$ time, which is optimal and beats the previous query time of $O(kn\alpha(n))$.


Champion spiders in the game of Graph Nim
In the game of Graph Nim, players take turns removing one or more edges incident
to a chosen vertex in a graph. The player that removes the last edge in the graph
wins. A spider graph is a champion if it has a SpragueGrundy number equal to the
number of edges in the graph. We investigate the the SpragueGrundy numbers of
various spider graphs when the number of paths or length of paths increase.


Manuscripts  
A note on approximate strengths of edges in a hypergraph
Let $H=(V,E)$ be an edgeweighted hypergraph of rank $r$. Kogan and Krauthgamer extended Benczúr and Karger's random sampling scheme for cut sparsification from graphs to hypergraphs. The sampling requires an algorithm for computing the approximate strengths of edges. In this note we extend the algorithm for graphs to hypergraphs and describe a nearlinear time algorithm to compute approximate strengths of edges; we build on a sparsification result for hypergraphs from our recent work. Combined with prior results we obtain faster algorithms for finding $(1+\epsilon)$approximate mincuts when the rank of the hypergraph is small.


Marking Streets to Improve Parking Density
Street parking spots for automobiles are a scarce commodity in most urban environments. The heterogeneity of car sizes makes it inefficient to rigidly define fixedsized spots. Instead, unmarked streets in cities like New York leave placement decisions to individual drivers, who have no direct incentive to maximize street utilization.
In this paper, we explore the effectiveness of two different behavioral interventions designed to encourage better parking, namely (1) educational campaigns to encourage parkers to "kiss the bumper" and reduce the distance between themselves and their neighbors, or (2) painting appropriatelyspaced markings on the street and urging drivers to "hit the line". Through analysis and simulation, we establish that the greatest densities are achieved when lines are painted to create spots roughly twice the length of averagesized cars. Kissthebumper campaigns are in principle more effective than hittheline for equal degrees of compliance, although we believe that the visual cues of painted lines induce better parking behavior.


Thesis  
Cuts and Connectivity in Graphs and Hypergraphs
In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs.
The main results are the following:
Coadvised by Karthik Chandrasekaran and Chandra Chekuri. 