About me
I'm a fourth year Ph.D. student at the Department of Computer Science at Princeton University. I work in Theoretical Computer Science and my advisor is Sanjeev Arora.
Before joining Princeton, I completed my undergraduate studies at the Computer Science and Engineering department at IIT Bombay and obtained a B.Tech. My advisor was Sundar Vishwanathan.
Research Interests
I'm broadly interested in theoretical computer science. My research interests include Algorithms, understanding the limits of Approximation, and Metric embeddings.
I have worked on questions around sparsest cut, fast SDP algorithms using the matrix exponential method, and metric embeddings.
My Curriculum Vitae.
Publications
- Approximating the Exponential,
the Lanczos Method and an \(\tilde{O}(m)\)-Time Spectral Algorithm for Balanced Separator
with Lorenzo Orecchia, Nisheeth K. Vishnoi. To appear in STOC 2012.
[arXiv] [Abstract]We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b \(\in (0,\frac{1}{2}],\) and a parameter \(\gamma,\) either finds an \(\Omega(b)\)-balanced cut of conductance \(O(\sqrt{\gamma})\) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least \(\gamma,\) and runs in time \(\tilde{O}(m).\) This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(-L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(-A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector v, in time roughly \(\tilde{O}(m_A),\) where \(m_A\) is the number of non-zero entries of A. This uses, in a non-trivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices in \(\tilde{O}(m_A)\) time. Finally, we prove that \(e^{-x}\) can be uniformly approximated up to a small additive error, in a non-negative interval [a,b] with a polynomial of degree roughly \(\sqrt{b-a}.\) While this result is of independent interest in approximation theory, we show that, via the Lanczos method from numerical analysis, it yields a simple algorithm to compute exp(-A)v for symmetric PSD matrices that runs in time roughly \(O(t_A\cdot \sqrt{\|A\|}),\) where \(t_A\) is time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance \(O(\sqrt{\gamma}),\) for balanced separator that runs in time \(\tilde{O}(\frac{m}{\sqrt{\gamma}}).\) This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.
- Finding Overlapping Communities in Social
Networks: Toward a Rigorous Approach
with Sanjeev Arora, Rong Ge, Grant Schoenebeck. To appear in EC 2012.
[arXiv] [Abstract]A community in a social network is usually understood to be a group of nodes more densely connected with each other than with the rest of the network. This is an important concept in most domains where networks arise: social, technological, biological, etc. For many years algorithms for finding communities implicitly assumed communities are nonoverlapping (leading to use of clustering-based approaches) but there is increasing interest in finding overlapping communities. A barrier to finding communities is that the solution concept is often defined in terms of an NP-complete problem such as Clique or Hierarchical Clustering. This paper seeks to initiate a rigorous approach to the problem of finding overlapping communities, where “rigorous” means that we clearly state the following: (a) the object sought by our algorithm (b) the assumptions about the underlying network (c) the (worst-case) running time. Our assumptions about the network lie between worst-case and average-case. An average-case analysis would require a precise probabilistic model of the network, on which there is currently no consensus. However, some plausible assumptions about network parameters can be gleaned from a long body of work in the sociology community spanning five decades focusing on the study of individual communities and ego-centric networks (in graph theoretic terms, this is the subgraph induced on a node’s neighborhood). Thus our assumptions are somewhat “local” in nature. Nevertheless they suffice to permit a rigorous analysis of running time of algorithms that recover global structure. Our algorithms use random sampling similar to that in property testing and algorithms for dense graphs. We note however that our networks are not necessarily dense graphs, not even in local neighborhoods. Our algorithms explore a local-global relationship between ego-centric and socio-centric networks that we hope will provide a fruitful framework for future work both in computer science and sociology.
- Nearly Optimal NP-Hardness of Vertex Cover on
\(k\)-Uniform \(k\)-Partite Hypergraphs
with Rishi Saket. In Approx 2011.
[arXiv] [Abstract]We study the problem of computing the minimum vertex cover on \(k\)-uniform \(k\)-partite hypergraphs when the \(k\)-partition is given. On bipartite graphs \((k=2)\), the minimum vertex cover can be computed in polynomial time. For general \(k\), the problem was studied by Lovász, who gave a \(\frac{k}{2}\)-approximation based on the standard LP relaxation. Subsequent work by Aharoni, Holzman and Krivelevich showed a tight integrality gap of \(\frac{k}{2} - o(1)\) for the LP relaxation. While this problem was known to be NP-hard for \(k\geq 3\), the first non-trivial NP-hardness of approximation factor of \(\frac{k}{4}-\epsilon\) was shown in a recent work by Guruswami and Saket. They also showed that assuming Khot's Unique Games Conjecture yields a \(\frac{k}{2}-\epsilon\) inapproximability for this problem, implying the optimality of Lovász's result.
In this work, we show that this problem is NP-hard to approximate within \(\frac{k}{2}-1+\frac{1}{2k}-\epsilon\). This hardness factor is off from the optimal by an additive constant of at most 1 for \(k\geq 4\). Our reduction relies on the
Multi-Layered PCP of Dinur et al. and uses a gadget -- based on biased Long Codes -- adapted from the LP integrality gap of Aharoni et al. The nature of our reduction requires the analysis of several Long Codes with different biases, for which we prove structural properties of the so called cross-intersecting collections of set families -- variants of which have been studied in extremal set theory. - Cuts in Cartesian Products of Graphs
with Madhur Tulsiani
[arXiv] [Abstract]The \(k\)-fold Cartesian product of a graph \(G\) is defined as a graph on tuples \((x_1,\ldots,x_k)\) where two tuples are connected if they form an edge in one of the positions and are equal in the rest. Starting with \(G\) as a single edge gives \(G^{\Box k}\) as a \(k\)-dimensional hypercube. We study the distributions of edges crossed by a cut in \(G^{\Box k}\) across the copies of \(G\) in different positions. This is a generalization of the notion of influences for cuts on the hypercube.
We show the analogues of the statements of Kahn, Kalai and Linial (KKL theorem) and that of Friedgut (Friedgut's Junta Theorem), for the setting of Cartesian products of arbitrary graphs. We also extends the work on studying isoperimetric constants for these graphs to the value of semidefinite relaxations for expansion. We connect the optimal values of the relaxations for computing expansion, given by various semidefinite hierarchies, for \(G\) and \(G^{\Box k}\).
- The Power of Weak v/s Strong Triangle Inequalities
with Mohammad Moharrami
[Abstract]In a recent paper, Lee and Moharrami construct a family of metrics \((X_n,d_n)\) with \(|X_n|=N\), such that \((X_n,\sqrt{d_n})\) embeds into \(\ell_2\) with constant distortion, but embedding \((X_n,d_n)\) into a metric of negative type requires distortion \(\Omega((\log N)^{\frac{1}{4}})\). In this paper, we build on their analysis, and improve their result by showing a \(\Omega((\log N)^{\frac{1}{3}})\) lower bound for embedding \((X_n,d_n)\) into a metric of negative type. Moreover, we show that this analysis is essentially tight by constructing a map that has distortion \(O((\log N)^{\frac{1}{3}})\).
This result implies a lower bound of \(\Omega((\log N)^{\frac{1}{3}})\) for the integrality gap of the relaxed version of Goemans-Linial semidefinite program with weak triangle inequalities for Non-uniform Sparsest Cut.
-
A Reformulation of the Arora-Rao-Vazirani Structure Theorem
with Sanjeev Arora, James Lee
[arXiv] [Abstract]In a well-known paper, Arora, Rao and Vazirani obtained an \(O(\log n)\) approximation to the Balanced Separator problem and Uniform Sparsest Cut. At the heart of their result is a geometric statement about sets of points that satisfy triangle inequalities, which also underlies subsequent work on approximation algorithms and geometric embeddings. In this note, we give an equivalent formulation of the Structure theorem in [ARV] in terms of the expansion of large sets in geometric graphs on sets of points satisfying triangle inequalities.
Contact
The best way to reach me is by email which would be
[last name] @ cs.princeton.edu
where [last name] is sachdeva