Suman Kalyan Bera

A depth-five lower bound for iterated matrix multiplication

We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require [EQUATION] gates when expressed as a homogeneous depth-five ΣΠΣΠΣ arithmetic circuit with the bottom fan-in bounded by N1/2-e. By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-four ΣΠΣΠ circuits. Our result extends a recent result of Kumar and Saraf, who gave the same [EQUATION] lower bound for homogeneous depth-four ΣΠΣΠ circuits computing IMM. It is analogous to a recent result of Kayal and Saha, who gave the same lower bound for homogeneous ΣΠΣΠΣ circuits (over characteristic zero) with bottom fan-in at most N1-e, for the harder problem of computing certain polynomials defined by Nisan--Wigderson designs.

Streaming quotient filter: a near optimal approximate duplicate detection approach for data streams

The unparalleled growth and popularity of the Internet coupled with the advent of diverse modern applications such as search engines, on-line transactions, climate warning systems, etc., has catered to an unprecedented expanse in the volume of data stored world-wide. Efficient storage, management, and processing of such massively exponential amount of data has emerged as a central theme of research in this direction. Detection and removal of redundancies and duplicates in real-time from such multi-trillion record-set to bolster resource and compute efficiency constitutes a challenging area of study. The infeasibility of storing the entire data from potentially unbounded data streams, with the need for precise elimination of duplicates calls for intelligent approximate duplicate detection algorithms. The literature hosts numerous works based on the well-known probabilistic bitmap structure, Bloom Filter and its variants. In this paper we propose a novel data structure, Streaming Quotient Filter, (SQF) for efficient detection and removal of duplicates in data streams. SQF intelligently stores the signatures of elements arriving on a data stream, and along with an eviction policy provides near zero false positive and false negative rates. We show that the near optimal performance of SQF is achieved with a very low memory requirement, making it ideal for real-time memory-efficient de-duplication applications having an extremely low false positive and false negative tolerance rates. We present detailed theoretical analysis of the working of SQF, providing a guarantee on its performance. Empirically, we compare SQF to alternate methods and show that the proposed method is superior in terms of memory and accuracy compared to the existing solutions. We also discuss Dynamic SQF for evolving streams and the parallel implementation of SQF.

Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams

We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles. For the important special case of counting triangles, we give a 4-pass, (1 +/- epsilon)-approximate, randomized algorithm using O-tilde(epsilon^(-2) m^(3/2) / T) space, where m is the number of edges and T is a promised lower bound on the number of triangles. This matches the space bound of a recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of Omega(min{m^(3/2)/T , m/sqrt(T)}), applicable at essentially all densities Omega(n) <= m <= O(n^2). We prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013). Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problems are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused. We use Turan graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary.

Approximation Algorithms for the Partition Vertex Cover Problem

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = (V,E), a cost function c: V → ℤ + , a partition P 1, …, P r of the edge set E, and a parameter k i for each partition P i . The goal is to find a minimum cost set of vertices which cover at least k i edges from the partition P i . We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(logr), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings.

Minimizing average flow-time under knapsack constraint

We give the first logarithmic approximation for minimizing average flow-time of jobs in the subset parallel machine setting (also called the restricted assignment setting) under a single knapsack constraint. In a knapsack constraint setting, each job has a profit, and the set of jobs which get scheduled must have a total profit of at least a quantity . Our result extends the work of Gupta et al. (2009) [8] who considered the special case where the profit of each job is unity. Our algorithm is based on rounding a natural LP relaxation for this problem. In fact, we show that one can use techniques based on iterative rounding.

Minimizing Average Flow-Time under Knapsack Constraint

We give the first logarithmic approximation for minimizing average flow-time of jobs in the subset parallel machine setting (also called the restricted assignment setting) under a single knapsack constraint. In a knapsack constraint setting, each job has a profit, and the set of jobs which get scheduled must have a total profit of at least a quantity Π. Our result extends the work of Gupta, Krishnaswamy, Kumar and Segev (APPROX 2009) who considered the special case where the profit of each job is unity. Our algorithm is based on rounding a natural LP relaxation for this problem. In fact, we show that one can use techniques based on iterative rounding.

Approximation algorithms for the partition vertex cover problem

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G = ( V , E ) , a cost function c : V ? Z + , a partition P 1 , ? , P r of the edge set E, and a parameter k i for each partition P i . The objective is to find a minimum cost set of vertices which cover at least k i edges from the partition P i . We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O ( log ? r ) , where r is the number of sets in the partition of the edge set. We also extend our result to more general settings. For example we consider a problem where additionally edges have profits, and we would like to pick a minimum cost set of vertices which cover edges of total profit at least ? i for each partition P i . We call this the Knapsack Partition Vertex Cover problem. We give an O ( log ? r ) approximation algorithm for this problem as well.