arXiv subjects

Barna Saha

Publications and source records attributed to Barna Saha.

At least 55 records · Page 3Linked to original sources

arXiv2020

Fair Correlation Clustering

In this paper we study the problem of correlation clustering under fairness constraints. In the classic correlation clustering problem, we are given a complete graph where each edge is labeled positive or negative. The goal is to obtain a clustering of the vertices that minimizes disagreements -- the number of negative edges trapped inside a cluster plus positive edges between different clusters. We consider two variations of fairness constraint for the problem of correlation clustering where each node has a color, and the goal is to form clusters that do not over-represent vertices of any color. The first variant aims to generate clusters with minimum disagreements, where the distribution of a feature (e.g. gender) in each cluster is same as the global distribution. For the case of two colors when the desired ratio of the number of colors in each cluster is $1:p$, we get $\mathcal{O}(p^2)$-approximation algorithm. Our algorithm could be extended to the case of multiple colors. We prove this problem is NP-hard. The second variant considers relative upper and lower bounds on the number of nodes of any color in a cluster. The goal is to avoid violating upper and lower bounds corresponding to each color in each cluster while minimizing the total number of disagreements. Along with our theoretical results, we show the effectiveness of our algorithm to generate fair clusters by empirical evaluation on real world data sets.

cs.DS↗

arXiv2019

Sublinear Algorithms for Gap Edit Distance

The edit distance is a way of quantifying how similar two strings are to one another by counting the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. A simple dynamic programming computes the edit distance between two strings of length $n$ in $O(n^2)$ time, and a more sophisticated algorithm runs in time $O(n+t^2)$ when the edit distance is $t$ [Landau, Myers and Schmidt, SICOMP 1998]. In pursuit of obtaining faster running time, the last couple of decades have seen a flurry of research on approximating edit distance, including polylogarithmic approximation in near-linear time [Andoni, Krauthgamer and Onak, FOCS 2010], and a constant-factor approximation in subquadratic time [Chakrabarty, Das, Goldenberg, Koucký and Saks, FOCS 2018]. We study sublinear-time algorithms for small edit distance, which was investigated extensively because of its numerous applications. Our main result is an algorithm for distinguishing whether the edit distance is at most $t$ or at least $t^2$ (the quadratic gap problem) in time $\tilde{O}(\frac{n}{t}+t^3)$. This time bound is sublinear roughly for all $t$ in $[ω(1), o(n^{1/3})]$, which was not known before. The best previous algorithms solve this problem in sublinear time only for $t=ω(n^{1/3})$ [Andoni and Onak, STOC 2009]. Our algorithm is based on a new approach that adaptively switches between uniform sampling and reading contiguous blocks of the input strings. In contrast, all previous algorithms choose which coordinates to query non-adaptively. Moreover, it can be extended to solve the $t$ vs $t^{2-ε}$ gap problem in time $\tilde{O}(\frac{n}{t^{1-ε}}+t^3)$.

cs.CC↗

arXiv2019

Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost

Several clustering frameworks with interactive (semi-supervised) queries have been studied in the past. Recently, clustering with same-cluster queries has become popular. An algorithm in this setting has access to an oracle with full knowledge of an optimal clustering, and the algorithm can ask the oracle queries of the form, "Does the optimal clustering put vertices $ u $ and $ v $ in the same cluster?" Due to its simplicity, this querying model can easily be implemented in real crowd-sourcing platforms and has attracted a lot of recent work. In this paper, we study the popular correlation clustering problem (Bansal et al., 2002) under this framework. Given a complete graph $G=(V,E)$ with positive and negative edge labels, correlation clustering objective aims to compute a graph clustering that minimizes the total number of disagreements, that is the negative intra-cluster edges and positive inter-cluster edges. Let $ C_{OPT} $ be the number of disagreements made by the optimal clustering. We present algorithms for correlation clustering whose error and query bounds are parameterized by $C_{OPT}$ rather than by the number of clusters. Indeed, a good clustering must have small $C_{OPT}$. Specifically, we present an efficient algorithm that recovers an exact optimal clustering using at most $2C_{OPT} $ queries and an efficient algorithm that outputs a $2$-approximation using at most $C_{OPT} $ queries. In addition, we show under a plausible complexity assumption, there does not exist any polynomial time algorithm that has an approximation ratio better than $1+α$ for an absolute constant $α>0$ with $o(C_{OPT})$ queries. We extensively evaluate our methods on several synthetic and real-world datasets using real crowd-sourced oracles. Moreover, we compare our approach against several known correlation clustering algorithms.

cs.DS↗

arXiv2019

Min-Max Correlation Clustering via MultiCut

Correlation clustering is a fundamental combinatorial optimization problem arising in many contexts and applications that has been the subject of dozens of papers in the literature. In this problem we are given a general weighted graph where each edge is labeled positive or negative. The goal is to obtain a partitioning (clustering) of the vertices that minimizes disagreements - weight of negative edges trapped inside a cluster plus positive edges between different clusters. Most of the papers on this topic mainly focus on minimizing total disagreement, a global objective for this problem. In this paper, we study a cluster-wise objective function that asks to minimize the maximum number of disagreements of each cluster, which we call min-max correlation clustering. The min-max objective is a natural objective that respects the quality of every cluster. In this paper, we provide the first nontrivial approximation algorithm for this problem achieving an $\mathcal{O}(\sqrt{\log n\cdot\max\{\log(|E^-|),\log(k)\}})$ approximation for general weighted graphs, where $|E^-|$ denotes the number of negative edges and $k$ is the number of clusters in the optimum solution. To do so, we also obtain a corresponding result for multicut where we wish to find a multicut solution while trying to minimize the total weight of cut edges on every component. The results are then further improved to obtain (i) $\mathcal{O}(r^2)$-approximation for min-max correlation clustering and min-max multicut for graphs that exclude $K_{r,r}$ minors (ii) a 14-approximation for the min-max correlation clustering on complete graphs.

cs.DS↗

arXiv2019

Paper Matching with Local Fairness Constraints

Automatically matching reviewers to papers is a crucial step of the peer review process for venues receiving thousands of submissions. Unfortunately, common paper matching algorithms often construct matchings suffering from two critical problems: (1) the group of reviewers assigned to a paper do not collectively possess sufficient expertise, and (2) reviewer workloads are highly skewed. In this paper, we propose a novel local fairness formulation of paper matching that directly addresses both of these issues. Since optimizing our formulation is not always tractable, we introduce two new algorithms, FairIR and FairFlow, for computing fair matchings that approximately optimize the new formulation. FairIR solves a relaxation of the local fairness formulation and then employs a rounding technique to construct a valid matching that provably maximizes the objective and only compromises on fairness with respect to reviewer loads and papers by a small constant. In contrast, FairFlow is not provably guaranteed to produce fair matchings, however it can be 2x as efficient as FairIR and an order of magnitude faster than matching algorithms that directly optimize for fairness. Empirically, we demonstrate that both FairIR and FairFlow improve fairness over standard matching algorithms on real conference data. Moreover, in comparison to state-of-the-art matching algorithms that optimize for fairness only, FairIR achieves higher objective scores, FairFlow achieves competitive fairness, and both are capable of more evenly allocating reviewers.

cs.DS↗

arXiv2019

Dynamic Set Cover: Improved Algorithms & Lower Bounds

We give new upper and lower bounds for the {\em dynamic} set cover problem. First, we give a $(1+ε) f$-approximation for fully dynamic set cover in $O(f^2\log n /ε^5)$ (amortized) update time, for any $ε> 0$, where $f$ is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to $O(f^2/ε^5)$, while still obtaining an $(1+ε) f$-approximation. These are the first algorithms that obtain an approximation factor linear in $f$ for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of $O(f^2)$ in the dynamic setting. To complement our upper bounds, we also show that a linear dependence of the update time on $f$ is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of $O(f^{1-ε})$ must have an approximation factor that is $Ω(n^δ)$ for some constant $δ>0$ under the Strong Exponential Time Hypothesis.

cs.DS↗

arXiv2018

The Geometric Block Model

To capture the inherent geometric features of many community detection problems, we propose to use a new random graph model of communities that we call a Geometric Block Model. The geometric block model generalizes the random geometric graphs in the same way that the well-studied stochastic block model generalizes the Erdos-Renyi random graphs. It is also a natural extension of random community models inspired by the recent theoretical and practical advancement in community detection. While being a topic of fundamental theoretical interest, our main contribution is to show that many practical community structures are better explained by the geometric block model. We also show that a simple triangle-counting algorithm to detect communities in the geometric block model is near-optimal. Indeed, even in the regime where the average degree of the graph grows only logarithmically with the number of vertices (sparse-graph), we show that this algorithm performs extremely well, both theoretically and practically. In contrast, the triangle-counting algorithm is far from being optimum for the stochastic block model. We simulate our results on both real and synthetic datasets to show superior performance of both the new model as well as our algorithm.

cs.SI↗

arXiv2017

A New Approximation Technique for Resource-Allocation Problems

We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys and Tardos on the generalized assignment problem to the setting where some jobs can be dropped. New concentration bounds for random bipartite matching are developed as well.

cs.DS↗

arXiv2017

Truly Sub-cubic Algorithms for Language Edit Distance and RNA Folding via Fast Bounded-Difference Min-Plus Product

It is a major open problem whether the $(\min,+)$-product of two $n\times n$ matrices has a truly sub-cubic (i.e. $O(n^{3-ε})$ for $ε>0$) time algorithm, in particular since it is equivalent to the famous All-Pairs-Shortest-Paths problem (APSP) in $n$-vertex graphs. Some restrictions of the $(\min,+)$-product to special types of matrices are known to admit truly sub-cubic algorithms, each giving rise to a special case of APSP that can be solved faster. In this paper we consider a new, different and powerful restriction in which all matrix entries are integers and one matrix can be arbitrary, as long as the other matrix has "bounded differences" in either its columns or rows, i.e. any two consecutive entries differ by only a small amount. We obtain the first truly sub-cubic algorithm for this bounded-difference $(\min,+)$-product (answering an open problem of Chan and Lewenstein). Our new algorithm, combined with a strengthening of an approach of L.~Valiant for solving context-free grammar parsing with matrix multiplication, yields the first truly sub-cubic algorithms for the following problems: Language Edit Distance (a major problem in the parsing community), RNA-folding (a major problem in bioinformatics) and Optimum Stack Generation (answering an open problem of Tarjan).

cs.DS↗

arXiv2017

Query Complexity of Clustering with Side Information

Suppose, we are given a set of $n$ elements to be clustered into $k$ (unknown) clusters, and an oracle/expert labeler that can interactively answer pair-wise queries of the form, "do two elements $u$ and $v$ belong to the same cluster?". The goal is to recover the optimum clustering by asking the minimum number of queries. In this paper, we initiate a rigorous theoretical study of this basic problem of query complexity of interactive clustering, and provide strong information theoretic lower bounds, as well as nearly matching upper bounds. Most clustering problems come with a similarity matrix, which is used by an automated process to cluster similar points together. Our main contribution in this paper is to show the dramatic power of side information aka similarity matrix on reducing the query complexity of clustering. A similarity matrix represents noisy pair-wise relationships such as one computed by some function on attributes of the elements. A natural noisy model is where similarity values are drawn independently from some arbitrary probability distribution $f_+$ when the underlying pair of elements belong to the same cluster, and from some $f_-$ otherwise. We show that given such a similarity matrix, the query complexity reduces drastically from $Θ(nk)$ (no similarity matrix) to $O(\frac{k^2\log{n}}{\cH^2(f_+\|f_-)})$ where $\cH^2$ denotes the squared Hellinger divergence. Moreover, this is also information-theoretic optimal within an $O(\log{n})$ factor. Our algorithms are all efficient, and parameter free, i.e., they work without any knowledge of $k, f_+$ and $f_-$, and only depend logarithmically with $n$. Along the way, our work also reveals intriguing connection to popular community detection models such as the {\em stochastic block model}, significantly generalizes them, and opens up many venues for interesting future research.

stat.ML↗

arXiv2017

Clustering with Noisy Queries

In this paper, we initiate a rigorous theoretical study of clustering with noisy queries (or a faulty oracle). Given a set of $n$ elements, our goal is to recover the true clustering by asking minimum number of pairwise queries to an oracle. Oracle can answer queries of the form : "do elements $u$ and $v$ belong to the same cluster?" -- the queries can be asked interactively (adaptive queries), or non-adaptively up-front, but its answer can be erroneous with probability $p$. In this paper, we provide the first information theoretic lower bound on the number of queries for clustering with noisy oracle in both situations. We design novel algorithms that closely match this query complexity lower bound, even when the number of clusters is unknown. Moreover, we design computationally efficient algorithms both for the adaptive and non-adaptive settings. The problem captures/generalizes multiple application scenarios. It is directly motivated by the growing body of work that use crowdsourcing for {\em entity resolution}, a fundamental and challenging data mining task aimed to identify all records in a database referring to the same entity. Here crowd represents the noisy oracle, and the number of queries directly relates to the cost of crowdsourcing. Another application comes from the problem of {\em sign edge prediction} in social network, where social interactions can be both positive and negative, and one must identify the sign of all pair-wise interactions by querying a few pairs. Furthermore, clustering with noisy oracle is intimately connected to correlation clustering, leading to improvement therein. Finally, it introduces a new direction of study in the popular {\em stochastic block model} where one has an incomplete stochastic block model matrix to recover the clusters.

stat.ML↗

arXiv2017

A Theoretical Analysis of First Heuristics of Crowdsourced Entity Resolution

Entity resolution (ER) is the task of identifying all records in a database that refer to the same underlying entity, and are therefore duplicates of each other. Due to inherent ambiguity of data representation and poor data quality, ER is a challenging task for any automated process. As a remedy, human-powered ER via crowdsourcing has become popular in recent years. Using crowd to answer queries is costly and time consuming. Furthermore, crowd-answers can often be faulty. Therefore, crowd-based ER methods aim to minimize human participation without sacrificing the quality and use a computer generated similarity matrix actively. While, some of these methods perform well in practice, no theoretical analysis exists for them, and further their worst case performances do not reflect the experimental findings. This creates a disparity in the understanding of the popular heuristics for this problem. In this paper, we make the first attempt to close this gap. We provide a thorough analysis of the prominent heuristic algorithms for crowd-based ER. We justify experimental observations with our analysis and information theoretic lower bounds.

cs.DB↗

arXiv2016

Clustering Via Crowdsourcing

In recent years, crowdsourcing, aka human aided computation has emerged as an effective platform for solving problems that are considered complex for machines alone. Using human is time-consuming and costly due to monetary compensations. Therefore, a crowd based algorithm must judiciously use any information computed through an automated process, and ask minimum number of questions to the crowd adaptively. One such problem which has received significant attention is {\em entity resolution}. Formally, we are given a graph $G=(V,E)$ with unknown edge set $E$ where $G$ is a union of $k$ (again unknown, but typically large $O(n^α)$, for $α>0$) disjoint cliques $G_i(V_i, E_i)$, $i =1, \dots, k$. The goal is to retrieve the sets $V_i$s by making minimum number of pair-wise queries $V \times V\to\{\pm1\}$ to an oracle (the crowd). When the answer to each query is correct, e.g. via resampling, then this reduces to finding connected components in a graph. On the other hand, when crowd answers may be incorrect, it corresponds to clustering over minimum number of noisy inputs. Even, with perfect answers, a simple lower and upper bound of $Θ(nk)$ on query complexity can be shown. A major contribution of this paper is to reduce the query complexity to linear or even sublinear in $n$ when mild side information is provided by a machine, and even in presence of crowd errors which are not correctable via resampling. We develop new information theoretic lower bounds on the query complexity of clustering with side information and errors, and our upper bounds closely match with them. Our algorithms are naturally parallelizable, and also give near-optimal bounds on the number of adaptive rounds required to match the query complexity.

cs.DS↗

arXiv2013

Distributed Data Placement via Graph Partitioning

With the widespread use of shared-nothing clusters of servers, there has been a proliferation of distributed object stores that offer high availability, reliability and enhanced performance for MapReduce-style workloads. However, relational workloads cannot always be evaluated efficiently using MapReduce without extensive data migrations, which cause network congestion and reduced query throughput. We study the problem of computing data placement strategies that minimize the data communication costs incurred by typical relational query workloads in a distributed setting. Our main contribution is a reduction of the data placement problem to the well-studied problem of {\sc Graph Partitioning}, which is NP-Hard but for which efficient approximation algorithms exist. The novelty and significance of this result lie in representing the communication cost exactly and using standard graphs instead of hypergraphs, which were used in prior work on data placement that optimized for different objectives (not communication cost). We study several practical extensions of the problem: with load balancing, with replication, with materialized views, and with complex query plans consisting of sequences of intermediate operations that may be computed on different servers. We provide integer linear programs (IPs) that may be used with any IP solver to find an optimal data placement. For the no-replication case, we use publicly available graph partitioning libraries (e.g., METIS) to efficiently compute nearly-optimal solutions. For the versions with replication, we introduce two heuristics that utilize the {\sc Graph Partitioning} solution of the no-replication case. Using the TPC-DS workload, it may take an IP solver weeks to compute an optimal data placement, whereas our reduction produces nearly-optimal solutions in seconds.

cs.DB↗

arXiv2013

Efficiently Computing Edit Distance to Dyck Language

Given a string $σ$ over alphabet $Σ$ and a grammar $G$ defined over the same alphabet, how many minimum number of repairs: insertions, deletions and substitutions are required to map $σ$ into a valid member of $G$ ? We investigate this basic question in this paper for $Dyck(s)$. $Dyck(s)$ is a fundamental context free grammar representing the language of well-balanced parentheses with s different types of parentheses and has played a pivotal role in the development of theory of context free languages. Computing edit distance to $Dyck(s)$ significantly generalizes string edit distance problem and has numerous applications ranging from repairing semi-structured documents such as XML to memory checking, automated compiler optimization, natural language processing etc. In this paper we give the first near-linear time algorithm for edit distance computation to $Dyck(s)$ that achieves a nontrivial approximation factor of $O(\frac{1}ε\log{OPT}(\log{n})^{\frac{1}ε})$ in $O(n^{1+ε}\log{n})$ time. In fact, given there exists an algorithm for computing string edit distance on input of size $n$ in $α(n)$ time with $β(n)$-approximation factor, we can devise an algorithm for edit distance problem to $Dyck(s)$ running in $\tilde{O}(n^{1+ε}+α(n))$ and achieving an approximation factor of $O(\frac{1}εβ(n)\log{OPT})$. We show that the framework for efficiently approximating edit distance to $Dyck(s)$ can be applied to many other languages. We illustrate this by considering various memory checking languages which comprise of valid transcripts of stacks, queues, priority queues, double-ended queues etc. Therefore, any language that can be recognized by these data structures, can also be repaired efficiently by our algorithm.

cs.DS↗

arXiv2013

A Constant Factor Approximation Algorithm for Fault-Tolerant k-Median

In this paper, we consider the fault-tolerant $k$-median problem and give the \emph{first} constant factor approximation algorithm for it. In the fault-tolerant generalization of classical $k$-median problem, each client $j$ needs to be assigned to at least $r_j \ge 1$ distinct open facilities. The service cost of $j$ is the sum of its distances to the $r_j$ facilities, and the $k$-median constraint restricts the number of open facilities to at most $k$. Previously, a constant factor was known only for the special case when all $r_j$s are the same, and a logarithmic approximation ratio for the general case. In addition, we present the first polynomial time algorithm for the fault-tolerant $k$-median problem on a path or a HST by showing that the corresponding LP always has an integral optimal solution. We also consider the fault-tolerant facility location problem, where the service cost of $j$ can be a weighted sum of its distance to the $r_j$ facilities. We give a simple constant factor approximation algorithm, generalizing several previous results which only work for nonincreasing weight vectors.

cs.DS↗

arXiv2012

AdCell: Ad Allocation in Cellular Networks

With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS- namely {\em AdCell}. In our model, a WSP charges the advertisers for showing their ads. Each advertiser has a valuation for specific types of customers in various times and locations and has a limit on the maximum available budget. Each query is in the form of time and location and is associated with one individual customer. In order to achieve a non-intrusive delivery, only a limited number of ads can be sent to each customer. Recently, new services have been introduced that offer location-based advertising over cellular network that fit in our model (e.g., ShopAlerts by AT&T) . We consider both online and offline version of the AdCell problem and develop approximation algorithms with constant competitive ratio. For the online version, we assume that the appearances of the queries follow a stochastic distribution and thus consider a Bayesian setting. Furthermore, queries may come from different distributions on different times. This model generalizes several previous advertising models such as online secretary problem \cite{HKP04}, online bipartite matching \cite{KVV90,FMMM09} and AdWords \cite{saberi05}. ...

cs.DS↗

arXiv2011

New Constructive Aspects of the Lovasz Local Lemma

The Lovász Local Lemma (LLL) states that the probability that none of a set of "bad" events happens is nonzero if the probability of each event is small compared to the number of bad events it depends on. A series of results have provided algorithms to efficiently construct structures whose existence is (non-constructively) guaranteed by the full asymmetric LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the Moser-Tardos procedure has sufficient randomness, leading to two classes of algorithmic applications. We first show that when an LLL application provides a small amount of slack, the running time of the Moser-Tardos algorithm is polynomial in the number of underlying independent variables (not events!), and can thus be used to give efficient constructions in cases where the underlying proof applies the LLL to super-polynomially many events (or where finding a bad event that holds is computationally hard). We demonstrate our method on applications including: the first constant-factor approximation algorithm for the Santa Claus problem, as well as efficient algorithms for acyclic edge coloring, non-repetitive graph colorings, and Ramsey-type graphs. Second, we show applications to cases where a few of the bad events can hold, leading to the first such algorithmic applications of the LLL: MAX $k$-SAT is an illustrative example of this.

cs.DS↗