arXiv subjects

Sapna Grover

Publications and source records attributed to Sapna Grover.

3 recordsLinked to original sources

arXiv2022

Constant factor approximations for Lower and Upper bounded Clusterings

Clustering is one of the most fundamental problem in Machine Learning. Researchers in the field often require a lower bound on the size of the clusters to maintain anonymity and upper bound for the ease of analysis. Specifying an optimal cluster size is a problem often faced by scientists. In this paper, we present a framework to obtain constant factor approximations for some prominent clustering objectives, with lower and upper bounds on cluster size. This enables scientists to give an approximate cluster size by specifying the lower and the upper bounds for it. Our results preserve the lower bounds but may violate the upper bound a little. %{GroverGD21_LBUBFL_Cocoon} to $2$. %namely, $k$ Center (LUkC) and $k$ Median (LUkM) problem. We study the problems when either of the bounds is uniform. We apply our framework to give the first constant factor approximations for LUkM and its generalization, $k$-facility location problem (LUkFL), with $\beta+1$ factor violation in upper bounds where $\beta$ is the violation of upper bounds in solutions of upper bounded $k$-median and $k$-facility location problems respectively. We also present a result on LUkC with uniform upper bounds and, its generalization, lower and (uniform) upper bounded $k$ supplier problem (LUkS). The approach also gives a result on lower and upper bounded facility location problem (LUFL), improving upon the upper bound violation of $5/2$ due to Gupta et al. We also reduce the violation in upper bounds for a special case when the gap between the lower and upper bounds is not too small.

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arXiv2021

First Approximation for Uniform Lower and Upper Bounded Facility Location Problem avoiding violation in Lower Bounds

With growing emphasis on e-commerce marketplace platforms where we have a central platform mediating between the seller and the buyer, it becomes important to keep a check on the availability and profitability of the central store. A store serving too less clients can be non-profitable and a store getting too many orders can lead to bad service to the customers which can be detrimental for the business. In this paper, we study the facility location problem(FL) with upper and lower bounds on the number of clients an open facility serves. Constant factor approximations are known for the restricted variants of the problem with only the upper bounds or only the lower bounds. The only work that deals with bounds on both the sides violates both the bounds [8]. In this paper, we present the first (constant factor) approximation for the problem violating the upper bound by a factor of (5/2) without violating the lower bounds when both the lower and the upper bounds are uniform. We first give a tri-criteria (constant factor) approximation violating both the upper and the lower bounds and then get rid of violation in lower bounds by transforming the problem instance to an instance of capacitated facility location problem.

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arXiv2016

Constant factor Approximation Algorithms for Uniform Hard Capacitated Facility Location Problems: Natural LP is not too bad

In this paper, we give first constant factor approximation for capacitated knapsack median problem (CKM) for hard uniform capacities, violating the budget only by an additive factor of $f_{max}$ where $f_{max}$ is the maximum cost of a facility opened by the optimal and violating capacities by $(2+\epsilon)$ factor. Natural LP for the problem is known to have an unbounded integrality gap when any one of the two constraints is allowed to be violated by a factor less than $2$. Thus, we present a result which is very close to the best achievable from the natural LP. To the best of our knowledge, the problem has not been studied earlier. For capacitated facility location problem with uniform capacities, a constant factor approximation algorithm is presented violating the capacities a little ($1 + \epsilon$). Though constant factor results are known for the problem without violating the capacities, the result is interesting as it is obtained by rounding the solution to the natural LP, which is known to have an unbounded integrality gap without violating the capacities. Thus, we achieve the best possible from the natural LP for the problem. The result shows that natural LP is not too bad. Finally, we raise some issues with the proofs of the results presented in \cite{capkmByrkaFRS2013} for capacitated $k$-facility location problem (C$k$FLP). \cite{capkmByrkaFRS2013} presents $O(1/\epsilon^2)$ approximation violating the capacities by a factor of $(2 + \epsilon)$ using dependent rounding. We first fix these issues using our techniques. Also, it can be argued that (deterministic) pipage rounding cannot be used to open the facilities instead of dependent rounding. Our techniques for CKM provide a constant factor approximation for CkFLP violating the capacities by $(2 + \epsilon)$.

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