arXiv subjects

Sayan Bhattacharya

Publications and source records attributed to Sayan Bhattacharya.

52 records · Page 3Linked to original sources

arXiv2019

New Amortized Cell-Probe Lower Bounds for Dynamic Problems

We build upon the recent papers by Weinstein and Yu (FOCS'16), Larsen (FOCS'12), and Clifford et al. (FOCS'15) to present a general framework that gives amortized lower bounds on the update and query times of dynamic data structures. Using our framework, we present two concrete results. (1) For the dynamic polynomial evaluation problem, where the polynomial is defined over a finite field of size $n^{1+Ω(1)}$ and has degree $n$, any dynamic data structure must either have an amortized update time of $Ω((\lg n/\lg \lg n)^2)$ or an amortized query time of $Ω((\lg n/\lg \lg n)^2)$. (2) For the dynamic online matrix vector multiplication problem, where we get an $n \times n$ matrix whose entires are drawn from a finite field of size $n^{Θ(1)}$, any dynamic data structure must either have an amortized update time of $Ω((\lg n/\lg \lg n)^2)$ or an amortized query time of $Ω(n \cdot (\lg n/\lg \lg n)^2)$. For these two problems, the previous works by Larsen (FOCS'12) and Clifford et al. (FOCS'15) gave the same lower bounds, but only for worst case update and query times. Our bounds match the highest unconditional lower bounds known till date for any dynamic problem in the cell-probe model.

cs.DS↗

arXiv2018

An Improved Algorithm for Incremental Cycle Detection and Topological Ordering in Sparse Graphs

We consider the problem of incremental cycle detection and topological ordering in a directed graph $G = (V, E)$ with $|V| = n$ nodes. In this setting, initially the edge-set $E$ of the graph is empty. Subsequently, at each time-step an edge gets inserted into $G$. After every edge-insertion, we have to report if the current graph contains a cycle, and as long as the graph remains acyclic, we have to maintain a topological ordering of the node-set $V$. Let $m$ be the total number of edges that get inserted into $G$. We present a randomized algorithm for this problem with $\tilde{O}(m^{4/3})$ total expected update time.

cs.DS↗

arXiv2018

Deterministically Maintaining a $(2+ε)$-Approximate Minimum Vertex Cover in $O(1/ε^2)$ Amortized Update Time

We consider the problem of maintaining an (approximately) minimum vertex cover in an $n$-node graph $G = (V, E)$ that is getting updated dynamically via a sequence of edge insertions/deletions. We show how to maintain a $(2+ε)$-approximate minimum vertex cover, "deterministically", in this setting in $O(1/ε^2)$ amortized update time. Prior to our work, the best known deterministic algorithm for maintaining a $(2+ε)$-approximate minimum vertex cover was due to Bhattacharya, Henzinger and Italiano [SODA 2015]. Their algorithm has an update time of $O(\log n/ε^2)$. Recently, Bhattacharya, Chakrabarty, Henzinger [IPCO 2017] and Gupta, Krishnaswamy, Kumar, Panigrahi [STOC 2017] showed how to maintain an $O(1)$-approximation in $O(1)$-amortized update time for the same problem. Our result gives an "exponential" improvement over the update time of Bhattacharya et al. [SODA 2015], and nearly matches the performance of the "randomized" algorithm of Solomon [FOCS 2016] who gets an approximation ratio of $2$ and an expected amortized update time of $O(1)$. We derive our result by analyzing, via a novel technique, a variant of the algorithm by Bhattacharya et al. We consider an idealized setting where the update time of an algorithm can take any arbitrary fractional value, and use insights from this setting to come up with an appropriate potential function. Conceptually, this framework mimics the idea of an LP-relaxation for an optimization problem. The difference is that instead of relaxing an integral objective function, we relax the update time of an algorithm itself. We believe that this technique will find further applications in the analysis of dynamic algorithms.

cs.DS↗

arXiv2017

Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for $(Δ+1)$- vertex coloring and $(2Δ-1)$-edge coloring in a graph with maximum degree $Δ$. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a $(Δ+1)$-vertex coloring with $O(\log Δ)$ expected amortized update time. (2) We present a deterministic algorithm which maintains a $(1+o(1))Δ$-vertex coloring with $O(\text{poly} \log Δ)$ amortized update time. (3) We present a simple, deterministic algorithm which maintains a $(2Δ-1)$-edge coloring with $O(\log Δ)$ worst-case update time. This improves the recent $O(Δ)$-edge coloring algorithm with $\tilde{O}(\sqrtΔ)$ worst-case update time by Barenboim and Maimon.

cs.DS↗

arXiv2017

Fully Dynamic Approximate Maximum Matching and Minimum Vertex Cover in $O(\log^3 n)$ Worst Case Update Time

We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data structure with approximation ratio $(3/2+ε)$ and worst case update time $O(m^{1/4}/ε^2)$, where $m$ is the number of edges in the graph. In recent past, Gupta and Peng [FOCS 2013] gave a deterministic data structure with approximation ratio $(1+ε)$ and worst case update time $O(\sqrt{m}/ε^2)$. No known randomised data structure beats the worst case update times of these two results. In contrast, the paper by Onak and Rubinfeld [STOC 2010] gave a randomised data structure with approximation ratio $O(1)$ and amortised update time $O(\log^2 n)$, where $n$ is the number of nodes in the graph. This was later improved by Baswana, Gupta and Sen [FOCS 2011] and Solomon [FOCS 2016], leading to a randomised date structure with approximation ratio $2$ and amortised update time $O(1)$. We bridge the polynomial gap between the worst case and amortised update times for this problem, without using any randomisation. We present a deterministic data structure with approximation ratio $(2+ε)$ and worst case update time $O(\log^3 n)$, for all sufficiently small constants $ε$.

cs.DS↗

arXiv2016

Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in $O(1)$ Amortized Update Time

We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of $2$ and an amortised update time of $O(1)$ with high probability. This algorithm requires the assumption of an {\em oblivious adversary}, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm's past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in $O(1)$ update time. In this paper, we resolve this question. We present a new {\em deterministic} fully dynamic algorithm that maintains a $O(1)$-approximate minimum vertex cover and maximum fractional matching, with an amortised update time of $O(1)$. Previously, the best deterministic algorithm for this problem was due to Bhattacharya, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of $(2+ε)$ and an amortised update time of $O(\log n/ε^2)$. Our results also extend to a fully dynamic $O(f^3)$-approximate algorithm with $O(f^2)$ amortized update time for the hypergraph vertex cover and fractional hypergraph matching problems, where every hyperedge has at most $f$ vertices.

cs.DS↗

arXiv2016

New Deterministic Approximation Algorithms for Fully Dynamic Matching

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a $(2+ε)$-approximate maximum matching in general graphs with $O(\text{poly}(\log n, 1/ε))$ update time. (2) An algorithm that maintains an $α_K$ approximation of the {\em value} of the maximum matching with $O(n^{2/K})$ update time in bipartite graphs, for every sufficiently large constant positive integer $K$. Here, $1\leq α_K < 2$ is a constant determined by the value of $K$. Result (1) is the first deterministic algorithm that can maintain an $o(\log n)$-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best {\em randomized} polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with {\em arbitrarily small polynomial} update time on bipartite graphs. Previously the best update time for this problem was $O(m^{1/4})$ [Bernstein et al. ICALP 2015], where $m$ is the current number of edges in the graph.

cs.DS↗

arXiv2016

Design of Dynamic Algorithms via Primal-Dual Method

We develop a dynamic version of the primal-dual method for optimization problems, and apply it to obtain the following results. (1) For the dynamic set-cover problem, we maintain an $O(f^2)$-approximately optimal solution in $O(f \cdot \log (m+n))$ amortized update time, where $f$ is the maximum "frequency" of an element, $n$ is the number of sets, and $m$ is the maximum number of elements in the universe at any point in time. (2) For the dynamic $b$-matching problem, we maintain an $O(1)$-approximately optimal solution in $O(\log^3 n)$ amortized update time, where $n$ is the number of nodes in the graph.

cs.DS↗

arXiv2015

Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams

While in many graph mining applications it is crucial to handle a stream of updates efficiently in terms of {\em both} time and space, not much was known about achieving such type of algorithm. In this paper we study this issue for a problem which lies at the core of many graph mining applications called {\em densest subgraph problem}. We develop an algorithm that achieves time- and space-efficiency for this problem simultaneously. It is one of the first of its kind for graph problems to the best of our knowledge. In a graph $G = (V, E)$, the "density" of a subgraph induced by a subset of nodes $S \subseteq V$ is defined as $|E(S)|/|S|$, where $E(S)$ is the set of edges in $E$ with both endpoints in $S$. In the densest subgraph problem, the goal is to find a subset of nodes that maximizes the density of the corresponding induced subgraph. For any $ε>0$, we present a dynamic algorithm that, with high probability, maintains a $(4+ε)$-approximation to the densest subgraph problem under a sequence of edge insertions and deletions in a graph with $n$ nodes. It uses $\tilde O(n)$ space, and has an amortized update time of $\tilde O(1)$ and a query time of $\tilde O(1)$. Here, $\tilde O$ hides a $O(\poly\log_{1+ε} n)$ term. The approximation ratio can be improved to $(2+ε)$ at the cost of increasing the query time to $\tilde O(n)$. It can be extended to a $(2+ε)$-approximation sublinear-time algorithm and a distributed-streaming algorithm. Our algorithm is the first streaming algorithm that can maintain the densest subgraph in {\em one pass}. The previously best algorithm in this setting required $O(\log n)$ passes [Bahmani, Kumar and Vassilvitskii, VLDB'12]. The space required by our algorithm is tight up to a polylogarithmic factor.

cs.DS↗

arXiv2015

Direct Observation of Re-entrant Multiferroic CuO at High Pressures

We have carried out a detailed experimental investigation on CuO using dielectric constant, ac resistance, Raman spectroscopy and X-ray diffraction measurements at high pressures and room temperature. Both dielectric constant and dielectric loss show an anomalous peak in the pressure range 3.4-4 GPa indicating a ferroelectric transition. Raman studies show anomalous behaviour of the Ag mode with a slope change in the mode frequency and a minimum in the mode FWHM at 3.4 GPa indicating a strong spin phonon coupling along [1 0 -1] direction. A step like behaviour in the intensity of the Ag mode is observed at 3.4 GPa, indicating a change in the polarization of the mode. A maximum in the intensity of (2,0,-2)Bragg peak at 3.4 GPa points the occurrence of critical scattering due to emergence of magnetic exchange interaction. All our experimental evidences show to the presence of re-entrant type-II multiferroic behaviour in CuO at about 4 GPa.

cond-mat.mtrl-sci↗

arXiv2014

Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph $G = (V,E)$, with $|V| = n$ and $|E| =m$, in $o(\sqrt{m}\,)$ time per update. In particular, for minimum vertex cover we provide deterministic data structures for maintaining a $(2+\eps)$ approximation in $O(\log n/\eps^2)$ amortized time per update. For maximum matching, we show how to maintain a $(3+\eps)$ approximation in $O(\min(\sqrt{n}/ε, m^{1/3}/\eps^2))$ {\em amortized} time per update, and a $(4+\eps)$ approximation in $O(m^{1/3}/\eps^2)$ {\em worst-case} time per update. Our data structure for fully dynamic minimum vertex cover is essentially near-optimal and settles an open problem by Onak and Rubinfeld from STOC' 2010.

cs.DS↗

arXiv2013

New Approximability Results for the Robust k-Median Problem

We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i \subseteq V}_{i=1}^m$. The objective is to open a set $F \subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i} \sum_{v \in S_i} d(F,v)$. Anthony et al.\ showed an $O(\log m)$ approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $Ω(\log m/ \log \log m)$ approximation hardness, unless ${\sf NP} \subseteq \bigcap_{δ>0} {\sf DTIME}(2^{n^δ})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.

cs.DS↗

arXiv2013

Constant-Competitive Prior-Free Auction with Ordered Bidders

A central problem in Microeconomics is to design auctions with good revenue properties. In this setting, the bidders' valuations for the items are private knowledge, but they are drawn from publicly known prior distributions. The goal is to find a truthful auction (no bidder can gain in utility by misreporting her valuation) that maximizes the expected revenue. Naturally, the optimal-auction is sensitive to the prior distributions. An intriguing question is to design a truthful auction that is oblivious to these priors, and yet manages to get a constant factor of the optimal revenue. Such auctions are called prior-free. Goldberg et al. presented a constant-approximate prior-free auction when there are identical copies of an item available in unlimited supply, bidders are unit-demand, and their valuations are drawn from i.i.d. distributions. The recent work of Leonardi et al. [STOC 2012] generalized this problem to non i.i.d. bidders, assuming that the auctioneer knows the ordering of their reserve prices. Leonardi et al. proposed a prior-free auction that achieves a $O(\log^* n)$ approximation. We improve upon this result, by giving the first prior-free auction with constant approximation guarantee.

cs.GT↗

arXiv2010

Budget Constrained Auctions with Heterogeneous Items

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian mechanism design contexts.

cs.GT↗

arXiv2009

On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids

We theoretically analyze the Cops and Robber Game for the first time in a multidimensional grid. It is shown that for an $n$-dimensional grid, at least $n$ cops are necessary to ensure capture of the robber. We also present a set of cop strategies for which $n$ cops are provably sufficient to catch the robber. Further, for two-dimensional grid, we provide an efficient cop strategy for which the robber is caught even by a single cop under certain conditions.

cs.DM↗

arXiv2009

Incentive Compatible Budget Elicitation in Multi-unit Auctions

In this paper, we consider the problem of designing incentive compatible auctions for multiple (homogeneous) units of a good, when bidders have private valuations and private budget constraints. When only the valuations are private and the budgets are public, Dobzinski {\em et al} show that the {\em adaptive clinching} auction is the unique incentive-compatible auction achieving Pareto-optimality. They further show thatthere is no deterministic Pareto-optimal auction with private budgets. Our main contribution is to show the following Budget Monotonicity property of this auction: When there is only one infinitely divisible good, a bidder cannot improve her utility by reporting a budget smaller than the truth. This implies that a randomized modification to the adaptive clinching auction is incentive compatible and Pareto-optimal with private budgets. The Budget Monotonicity property also implies other improved results in this context. For revenue maximization, the same auction improves the best-known competitive ratio due to Abrams by a factor of 4, and asymptotically approaches the performance of the optimal single-price auction. Finally, we consider the problem of revenue maximization (or social welfare) in a Bayesian setting. We allow the bidders have public size constraints (on the amount of good they are willing to buy) in addition to private budget constraints. We show a simple poly-time computable 5.83-approximation to the optimal Bayesian incentive compatible mechanism, that is implementable in dominant strategies. Our technique again crucially needs the ability to prevent bidders from over-reporting budgets via randomization.

cs.GT↗