I'm a research scientist at Yahoo! Research in New York. My research interests are algorithms, combinatorial optimization and computational geometry. I'm also interested in more applied problems with nice theoretical components. Here is my CV(last updated Oct 2018) and old research statement.
I obtained my PhD in Computer Science from University of Illinois at Urbana-Champaign in May 2018. My advisors were Karthik Chandrasekaran and Chandra Chekuri. I've previously held a summer visiting position at National Institute of Informatics, hosted by Ken-ichi Kawarabayashi, and New York University, hosted by Boris Aronov. I finished my undergrad in mathematics at Stony Brook University in 2013, and spent Spring 2012 in Budapest Semesters in Mathematics.
You can contact me through email email@example.com.
Karger used spanning tree packings to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm. Thorup developed a fast deterministic algorithm for the minimum -cut problem via greedy recursive tree packings. In this paper we revisit properties of an LP relaxation for -cut proposed by Naor and Rabani, and analyzed by Chekuri, Guha and Naor. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the -cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of . We also improve the bound on the number of -approximate -cuts. Second, we give a simple proof that the integrality gap of the LP is . Third, we show that an optimum solution to the LP relaxation, for all values of , is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona and Ravi and Sinha; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup.
In the hypergraph -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 connected components. This problem is known to be at least as hard as the densest -subgraph problem when k is part of the input (Chekuri-Li, 2015). We present a randomized polynomial time algorithm to solve the hypergraph -cut problem for constant . Our algorithm solves the more general hedge -cut problem when the subgraph induced by every hedge has a constant number of connected components. In the hedge -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 . 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.
The computational complexity of multicut-like 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.
Our techniques for the algorithms are combinatorial, based on LPs and based on enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.
Given a multiset of positive integers and a target integer , the subset sum problem is to decide if there is a subset of that sums up to . We present a new divide-and-conquer algorithm that computes all the realizable subset sums up to an integer in , where is the sum of all elements in and hides polylogarithmic factors. This result improves upon the standard dynamic programming algorithm that runs in 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 time, where m is the order of the group.
We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph with , and the best known algorithm to compute a global minimum cut in runs in time for the uncapacitated case and in time for the capacitated case. We show the following new results.
The notion of element-connectivity has found several important applications in network design and routing problems. We focus on a reduction step that preserves the element-connectivity, which when applied repeatedly allows one to reduce the original graph to a simpler one. This pre-processing 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 element-connectivity including the problem of achieving the aforementioned graph simplification. We utilize the underlying submodularity properties of element-connectivity to derive faster algorithms.
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 time, improving an earlier -time algorithm of Cortese et al.. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in time. We also describe algorithms that detect weak simplicity in time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.
We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph with , and the fastest known algorithm to compute a global minimum cut in runs in time for the uncapacitated case, and in time for the capacitated case. We show the following new results.
Our results build upon properties of vertex orderings that were inspired by the maximum adjacency ordering for graphs due to Nagamochi and Ibaraki. Unlike graphs we observe that there are several orderings for hypergraphs and these yield different insights.
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 , where and are the input size and the output size of the algorithm, respectively.
In the fixed-terminal bicut problem, the input is a directed graph with two specified nodes and and the goal is to find a smallest subset of edges whose removal ensures that cannot reach and cannot reach . 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 and such that cannot reach and cannot reach . Fixed-terminal bicut and global bicut are natural extensions of -min cut and global min-cut respectively, from undirected graphs to directed graphs. Fixed-terminal bicut is NP-hard, admits a simple -approximation, and does not admit a -approximation for any constant assuming the unique games conjecture. In this work, we show that global bicut admits a -approximation, thus improving on the approximability of the global variant in comparison to the fixed-terminal variant.
For a simple undirected graph with vertices and edges, we consider a data structure that given a query of a pair of vertices , and an integer , it returns edge-disjoint -paths. The data structure takes time to build, using space, and each query takes time, which is optimal and beats the previous query time of .
Congr. Numer. .
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 Sprague-Grundy number equal to the number of edges in the graph. We investigate the the Sprague-Grundy numbers of various spider graphs when the number of paths or length of paths increase.
The capacitated vehicle routing problem (CVRP) is one of the most well known NP-hard combinatorial optimization problem. Single depot CVRP with general metric is NP-hard even for fixed capacity , while polynomial time solvable for fixed capacity . We consider the variant of CVRP where restocking is available. We show that if there is a constant number of depots, then the problem can be solved in polynomial time when . More generally, when there are depots without a starting vehicle, there is a polynomial time algorithm for constant .
Subset Sum is a classical optimization problem taught to undergraduates as an example of an NP-hard problem, which is amenable to dynamic programming, yielding polynomial running time if the input numbers are relatively small. Formally, given a set of positive integers and a target integer , the Subset Sum problem is to decide if there is a subset of that sums up to . Dynamic programming yields an algorithm with running time . Recently, the authors [Koiliaris & Xu SODA '17] improved the running time to , and it was further improved to by a somewhat involved randomized algorithm by Bringmann [Bringmann SODA '17], where hides polylogarithmic factors. Here, we present a new and significantly simpler algorithm with running time . While not the fastest, we believe the new algorithm and analysis are simple enough to be presented in an algorithms class, as a striking example of a divide-and-conquer algorithm that uses FFT to a problem that seems (at first) unrelated. In particular, the algorithm and its analysis can be described in full detail in two pages (see pages 3-5).
Street parking spots for automobiles are a scarce commodity in most urban environments. The heterogeneity of car sizes makes it inefficient to rigidly define fixed-sized 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 appropriately-spaced 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 average-sized cars. Kiss-the-bumper campaigns are in principle more effective than hit-the-line for equal degrees of compliance, although we believe that the visual cues of painted lines induce better parking behavior.
In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following: