arXiv subjects

Deeparnab Chakrabarty

Publications and source records attributed to Deeparnab Chakrabarty.

At least 37 records · Page 2Linked to original sources

arXiv2021

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making $\mathrm{poly}(N)$ queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where $N$ denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least $N$ rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with $\mathrm{poly}(N)$ queries using only poly-logarithmic rounds of adaptivity. An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making $\mathrm{poly}(N)$ queries necessarily requires $Ω(\log N/\log \log N)$ rounds. This left open the possibility of efficient SFM algorithms in poly-logarithmic rounds. For matroid intersection, even the possibility of a constant round, $\mathrm{poly}(N)$ query algorithm was not hitherto ruled out. In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making $\mathrm{poly}(N)$ queries requires $\tildeΩ\left(N^{1/3}\right)$ rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most $2^{N^{1-δ}}$ queries, for any constant $δ> 0$. Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model.

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arXiv2021

Graph Connectivity and Single Element Recovery via Linear and OR Queries

We study the problem of finding a spanning forest in an undirected, $n$-vertex multi-graph under two basic query models. One is the Linear query model which are linear measurements on the incidence vector induced by the edges; the other is the weaker OR query model which only reveals whether a given subset of plausible edges is empty or not. At the heart of our study lies a fundamental problem which we call the {\em single element recovery} problem: given a non-negative real vector $x$ in $N$ dimension, return a single element $x_j > 0$ from the support. Queries can be made in rounds, and our goals is to understand the trade-offs between the query complexity and the rounds of adaptivity needed to solve these problems, for both deterministic and randomized algorithms. These questions have connections and ramifications to multiple areas such as sketching, streaming, graph reconstruction, and compressed sensing. Our main results are: * For the single element recovery problem, it is easy to obtain a deterministic, $r$-round algorithm which makes $(N^{1/r}-1)$-queries per-round. We prove that this is tight: any $r$-round deterministic algorithm must make $\geq (N^{1/r} - 1)$ linear queries in some round. In contrast, a $1$-round $O(\log^2 N)$-query randomized algorithm which succeeds 99% of the time is known to exist. * We design a deterministic $O(r)$-round, $\tilde{O}(n^{1+1/r})$-OR query algorithm for graph connectivity. We complement this with an $\tildeΩ(n^{1 + 1/r})$-lower bound for any $r$-round deterministic algorithm in the OR-model. * We design a randomized, $2$-round algorithm for the graph connectivity problem which makes $\tilde{O}(n)$-OR queries. In contrast, we prove that any $1$-round algorithm (possibly randomized) requires $\tildeΩ(n^2)$-OR queries.

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arXiv2021

Better Algorithms for Individually Fair $k$-Clustering

We study data clustering problems with $\ell_p$-norm objectives (e.g. $k$-Median and $k$-Means) in the context of individual fairness. The dataset consists of $n$ points, and we want to find $k$ centers such that (a) the objective is minimized, while (b) respecting the individual fairness constraint that every point $v$ has a center within a distance at most $r(v)$, where $r(v)$ is $v$'s distance to its $(n/k)$th nearest point. Jung, Kannan, and Lutz [FORC 2020] introduced this concept and designed a clustering algorithm with provable (approximate) fairness and objective guarantees for the $\ell_\infty$ or $k$-Center objective. Mahabadi and Vakilian [ICML 2020] revisited this problem to give a local-search algorithm for all $\ell_p$-norms. Empirically, their algorithms outperform Jung et. al.'s by a large margin in terms of cost (for $k$-Median and $k$-Means), but they incur a reasonable loss in fairness. In this paper, our main contribution is to use Linear Programming (LP) techniques to obtain better algorithms for this problem, both in theory and in practice. We prove that by modifying known LP rounding techniques, one gets a worst-case guarantee on the objective which is much better than in MV20, and empirically, this objective is extremely close to the optimal. Furthermore, our theoretical fairness guarantees are comparable with MV20 in theory, and empirically, we obtain noticeably fairer solutions. Although solving the LP {\em exactly} might be prohibitive, we demonstrate that in practice, a simple sparsification technique drastically improves the run-time of our algorithm.

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arXiv2021

Robust $k$-Center with Two Types of Radii

In the non-uniform $k$-center problem, the objective is to cover points in a metric space with specified number of balls of different radii. Chakrabarty, Goyal, and Krishnaswamy [ICALP 2016, Trans. on Algs. 2020] (CGK, henceforth) give a constant factor approximation when there are two types of radii. In this paper, we give a constant factor approximation for the two radii case in the presence of outliers. To achieve this, we need to bypass the technical barrier of bad integrality gaps in the CGK approach. We do so using "the ellipsoid method inside the ellipsoid method": use an outer layer of the ellipsoid method to reduce to stylized instances and use an inner layer of the ellipsoid method to solve these specialized instances. This idea is of independent interest and could be applicable to other problems. Keywords: Approximation, Clustering, Outliers, and Round-or-Cut.

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arXiv2019

Domain Reduction for Monotonicity Testing: A $o(d)$ Tester for Boolean Functions in $d$-Dimensions

We describe a $\tilde{O}(d^{5/6})$-query monotonicity tester for Boolean functions $f:[n]^d \to \{0,1\}$ on the $n$-hypergrid. This is the first $o(d)$ monotonicity tester with query complexity independent of $n$. Motivated by this independence of $n$, we initiate the study of monotonicity testing of measurable Boolean functions $f:\mathbb{R}^d \to \{0,1\}$ over the continuous domain, where the distance is measured with respect to a product distribution over $\mathbb{R}^d$. We give a $\tilde{O}(d^{5/6})$-query monotonicity tester for such functions. Our main technical result is a domain reduction theorem for monotonicity. For any function $f:[n]^d \to \{0,1\}$, let $ε_f$ be its distance to monotonicity. Consider the restriction $\hat{f}$ of the function on a random $[k]^d$ sub-hypergrid of the original domain. We show that for $k = \text{poly}(d/ε)$, the expected distance of the restriction is $\mathbb{E}[ε_{\hat{f}}] = Ω(ε_f)$. Previously, such a result was only known for $d=1$ (Berman-Raskhodnikova-Yaroslavtsev, STOC 2014). Our result for testing Boolean functions over $[n]^d$ then follows by applying the $d^{5/6}\cdot \text{poly}(1/ε,\log n, \log d)$-query hypergrid tester of Black-Chakrabarty-Seshadhri (SODA 2018). To obtain the result for testing Boolean functions over $\mathbb{R}^d$, we use standard measure theoretic tools to reduce monotonicity testing of a measurable function $f$ to monotonicity testing of a discretized version of $f$ over a hypergrid domain $[N]^d$ for large, but finite, $N$ (that may depend on $f$). The independence of $N$ in the hypergrid tester is crucial to getting the final tester over $\mathbb{R}^d$.

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arXiv2019

Faster Matroid Intersection

In this paper we consider the classic matroid intersection problem: given two matroids $\M_{1}=(V,\I_{1})$ and $\M_{2}=(V,\I_{2})$ defined over a common ground set $V$, compute a set $S\in\I_{1}\cap\I_{2}$ of largest possible cardinality, denoted by $r$. We consider this problem both in the setting where each $\M_{i}$ is accessed through an independence oracle, i.e. a routine which returns whether or not a set $S\in\I_{i}$ in $\indep$ time, and the setting where each $\M_{i}$ is accessed through a rank oracle, i.e. a routine which returns the size of the largest independent subset of $S$ in $\M_{i}$ in $\rank$ time. In each setting we provide faster exact and approximate algorithms. Given an independence oracle, we provide an exact $O(nr\log r \indep)$ time algorithm. This improves upon the running time of $O(nr^{1.5} \indep)$ due to Cunningham in 1986 and $\tilde{O}(n^{2} \indep+n^{3})$ due to Lee, Sidford, and Wong in 2015. We also provide two algorithms which compute a $(1-ε)$-approximate solution to matroid intersection running in times $\tilde{O}(n^{1.5}/\eps^{1.5} \indep)$ and $\tilde{O}((n^{2}r^{-1}ε^{-2}+r^{1.5}ε^{-4.5}) \indep)$, respectively. These results improve upon the $O(nr/\eps \indep)$-time algorithm of Cunningham as noted recently by Chekuri and Quanrud. Given a rank oracle, we provide algorithms with even better dependence on $n$ and $r$. We provide an $O(n\sqrt{r}\log n \rank)$-time exact algorithm and an $O(nε^{-1}\log n \rank)$-time algorithm which obtains a $(1-\eps)$-approximation to the matroid intersection problem. The former result improves over the $\tilde{O}(nr \rankt+n^{3})$-time algorithm by Lee, Sidford, and Wong. The rank oracle is of particular interest as the matroid intersection problem with this oracle is a special case of the submodular function minimization problem with an evaluation oracle.

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arXiv2019

On a Decentralized $(Δ{+}1)$-Graph Coloring Algorithm

We consider a decentralized graph coloring model where each vertex only knows its own color and whether some neighbor has the same color as it. The networking community has studied this model extensively due to its applications to channel selection, rate adaptation, etc. Here, we analyze variants of a simple algorithm of Bhartia et al. [Proc., ACM MOBIHOC, 2016]. In particular, we introduce a variant which requires only $O(n\logΔ)$ expected recolorings that generalizes the coupon collector problem. Finally, we show that the $O(nΔ)$ bound Bhartia et al. achieve for their algorithm still holds and is tight in adversarial scenarios.

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arXiv2019

Fair Algorithms for Clustering

We study the problem of finding low-cost Fair Clusterings in data where each data point may belong to many protected groups. Our work significantly generalizes the seminal work of Chierichetti et.al. (NIPS 2017) as follows. - We allow the user to specify the parameters that define fair representation. More precisely, these parameters define the maximum over- and minimum under-representation of any group in any cluster. - Our clustering algorithm works on any $\ell_p$-norm objective (e.g. $k$-means, $k$-median, and $k$-center). Indeed, our algorithm transforms any vanilla clustering solution into a fair one incurring only a slight loss in quality. - Our algorithm also allows individuals to lie in multiple protected groups. In other words, we do not need the protected groups to partition the data and we can maintain fairness across different groups simultaneously. Our experiments show that on established data sets, our algorithm performs much better in practice than what our theoretical results suggest.

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arXiv2019

Simpler and Better Algorithms for Minimum-Norm Load Balancing

Recently, Chakrabarty and Swamy (STOC 2019) introduced the {\em minimum-norm load-balancing} problem on unrelated machines, wherein we are given a set $J$ of jobs that need to be scheduled on a set of $m$ unrelated machines, and a monotone, symmetric norm; We seek an assignment $\sg:J\mapsto[m]$ that minimizes the norm of the resulting load vector $\lvec_\sg\in\R_+^m$, where $\lvec_\sg(i)$ is the load on machine $i$ under the assignment $\sg$. Besides capturing all $\ell_p$ norms, symmetric norms also capture other norms of interest including top-$\ell$ norms, and ordered norms. Chakrabarty and Swamy (STOC 2019) give a $(38+\ve)$-approximation algorithm for this problem via a general framework they develop for minimum-norm optimization that proceeds by first carefully reducing this problem (in a series of steps) to a problem called \minmax ordered load balancing, and then devising a so-called deterministic oblivious LP-rounding algorithm for ordered load balancing. We give a direct, and simple $4$-approximation algorithm for the minimum-norm load balancing based on rounding a (near-optimal) solution to a novel convex-programming relaxation for the problem. Whereas the natural convex program encoding minimum-norm load balancing problem has a large non-constant integrality gap, we show that this issue can be remedied by including a key constraint that bounds the "norm of the job-cost vector." Our techniques also yield a (essentially) $4$-approximation for: (a) {\em multi-norm load balancing}, wherein we are given multiple monotone symmetric norms, and we seek an assignment respecting a given budget for each norm; (b) the best {\em simultaneous approximation factor} achievable for all symmetric norms for a given instance.

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arXiv2018

Approximation Algorithms for Minimum Norm and Ordered Optimization Problems

$ $In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In $k$-clustering, opening $k$ facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : Given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in the unrelated machine load balancing and $k$-clustering setting. Our concrete results are the following. $\bullet$ We give constant factor approximation algorithms for the minimum norm load balancing problem in unrelated machines, and the minimum norm $k$-clustering problem. To our knowledge, our results constitute the first constant-factor approximations for such a general suite of objectives. $\bullet$ In load balancing with unrelated machines, we give a $2$-approximation for the problem of finding an assignment minimizing the sum of the largest $\ell$ loads, for any $\ell$. We give a $(2+\varepsilon)$-approximation for the so-called ordered load-balancing problem. $\bullet$ For $k$-clustering, we give a $(5+\varepsilon)$-approximation for the ordered $k$-median problem significantly improving the constant factor approximations from Byrka, Sornat, and Spoerhase (STOC 2018) and Chakrabarty and Swamy (ICALP 2018). $\bullet$ Our techniques also imply $O(1)$ approximations to the best simultaneous optimization factor for any instance of the unrelated machine load-balancing and the $k$-clustering setting. To our knowledge, these are the first positive simultaneous optimization results in these settings.

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arXiv2018

Generalized Center Problems with Outliers

We study the $\mathcal{F}$-center problem with outliers: given a metric space $(X,d)$, a general down-closed family $\mathcal{F}$ of subsets of $X$, and a parameter $m$, we need to locate a subset $S\in \mathcal{F}$ of centers such that the maximum distance among the closest $m$ points in $X$ to $S$ is minimized. Our main result is a dichotomy theorem. Colloquially, we prove that there is an efficient $3$-approximation for the $\mathcal{F}$-center problem with outliers if and only if we can efficiently optimize a poly-bounded linear function over $\mathcal{F}$ subject to a partition constraint. One concrete upshot of our result is a polynomial time $3$-approximation for the knapsack center problem with outliers for which no (true) approximation algorithm was known.

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arXiv2018

Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling

Given a non-negative $n \times m$ real matrix $A$, the {\em matrix scaling} problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified target value for it. This problem arises in many algorithmic applications, perhaps most notably as a preconditioning step in solving a linear system of equations. One of the most natural and by now classical approach to matrix scaling is the Sinkhorn-Knopp algorithm (also known as the RAS method) where one alternately scales either all rows or all columns to meet the target values. In addition to being extremely simple and natural, another appeal of this procedure is that it easily lends itself to parallelization. A central question is to understand the rate of convergence of the Sinkhorn-Knopp algorithm. In this paper, we present an elementary convergence analysis for the Sinkhorn-Knopp algorithm that improves upon the previous best bound. In a nutshell, our approach is to show a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold $δ$, and then connect the KL-divergence with $\ell_1$ and $\ell_2$ distances. For $\ell_1$, we can use Pinsker's inequality. For $\ell_2$, we develop a strengthening of Pinsker's inequality, called (KL vs $\ell_1/\ell_2$) in the paper, which lower bounds the KL-divergence by a combination of $\ell_1$ and $\ell_2$ distance. This inequality may be of independent interest. The idea of studying Sinkhorn-Knopp convergence via KL-divergence is not new and has indeed been previously explored. Our contribution is an elementary, self-contained presentation of this approach and an interesting new inequality that yields a significantly stronger convergence guarantee for the extensively studied $\ell_2$-error.

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arXiv2018

Adaptive Boolean Monotonicity Testing in Total Influence Time

The problem of testing monotonicity of a Boolean function $f:\{0,1\}^n \to \{0,1\}$ has received much attention recently. Denoting the proximity parameter by $\varepsilon$, the best tester is the non-adaptive $\widetilde{O}(\sqrt{n}/\varepsilon^2)$ tester of Khot-Minzer-Safra (FOCS 2015). Let $I(f)$ denote the total influence of $f$. We give an adaptive tester whose running time is $I(f)poly(\varepsilon^{-1}\log n)$.

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arXiv2017

Interpolating between $k$-Median and $k$-Center: Approximation Algorithms for Ordered $k$-Median

We consider a generalization of $k$-median and $k$-center, called the {\em ordered $k$-median} problem. In this problem, we are given a metric space $(\mathcal{D},\{c_{ij}\})$ with $n=|\mathcal{D}|$ points, and a non-increasing weight vector $w\in\mathbb{R}_+^n$, and the goal is to open $k$ centers and assign each point each point $j\in\mathcal{D}$ to a center so as to minimize $w_1\cdot\text{(largest assignment cost)}+w_2\cdot\text{(second-largest assignment cost)}+\ldots+w_n\cdot\text{($n$-th largest assignment cost)}$. We give an $(18+ε)$-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of $\{0,1\}$-weights, which models the problem of minimizing the $\ell$ largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) $k$-median problem showing that LP-relative guarantees for $k$-median translate to guarantees for the ordered $k$-median problem; this yields a nice and clean $(8.5+ε)$-approximation algorithm for $\{0,1\}$ weights.

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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.

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arXiv2017

A $o(d) \cdot \text{polylog}~n$ Monotonicity Tester for Boolean Functions over the Hypergrid $[n]^d$

We study monotonicity testing of Boolean functions over the hypergrid $[n]^d$ and design a non-adaptive tester with $1$-sided error whose query complexity is $\tilde{O}(d^{5/6})\cdot \text{poly}(\log n,1/ε)$. Previous to our work, the best known testers had query complexity linear in $d$ but independent of $n$. We improve upon these testers as long as $n = 2^{d^{o(1)}}$. To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.

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arXiv2017

A Lower Bound for Nonadaptive, One-Sided Error Testing of Unateness of Boolean Functions over the Hypercube

A Boolean function $f:\{0,1\}^d \mapsto \{0,1\}$ is unate if, along each coordinate, the function is either nondecreasing or nonincreasing. In this note, we prove that any nonadaptive, one-sided error unateness tester must make $Ω(\frac{d}{\log d})$ queries. This result improves upon the $Ω(\frac{d}{\log^2 d})$ lower bound for the same class of testers due to Chen et al. (STOC, 2017).

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arXiv2017

Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps

We study the problem of testing unateness of functions $f:\{0,1\}^d \to \mathbb{R}.$ We give a $O(\frac{d}ε \cdot \log\frac{d}ε)$-query nonadaptive tester and a $O(\frac{d}ε)$-query adaptive tester and show that both testers are optimal for a fixed distance parameter $ε$. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. We also generalize our results to obtain optimal unateness testers for functions $f:[n]^d \to \mathbb{R}$. Our results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form $\{0,1\}^d$ and, more generally, $[n]^d$. This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions $f:[n]^d \to \mathbb{R}$.

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