arXiv subjects

Binay Bhattacharya

Publications and source records attributed to Binay Bhattacharya.

12 recordsLinked to original sources

arXiv2023

A Subquadratic Time Algorithm for the Weighted $k$-Center Problem on Cactus Graphs

The weighted $k$-center problem in graphs is a classical facility location problem where we place $k$ centers on the graph, which minimize the maximum weighted distance of a vertex to its nearest center. We study this problem when the underlying graph is a cactus with $n$ vertices and present an $O(n \log^2 n)$ time algorithm for the same. This time complexity improves upon the $O(n^2)$ time algorithm by Ben-Moshe et al. [TCS 2007], which is the current state-of-the-art.

cs.DS↗

arXiv2022

An Efficient Algorithm for the Proximity Connected Two Center Problem

Given a set $P$ of $n$ points in the plane, the $k$-center problem is to find $k$ congruent disks of minimum possible radius such that their union covers all the points in $P$. The $2$-center problem is a special case of the $k$-center problem that has been extensively studied in the recent past \cite{CAHN,HT,SH}. In this paper, we consider a generalized version of the $2$-center problem called \textit{proximity connected} $2$-center (PCTC) problem. In this problem, we are also given a parameter $δ\geq 0$ and we have the additional constraint that the distance between the centers of the disks should be at most $δ$. Note that when $δ=0$, the PCTC problem is reduced to the $1$-center(minimum enclosing disk) problem and when $δ$ tends to infinity, it is reduced to the $2$-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 \cite{ACN0}, but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic $O(n^2\log n)$ time algorithm for the problem.

cs.CG↗

arXiv2019

Computation of Spatial Skyline Points

We discuss a method of finding skyline or non-dominated sites in a set $P$ of $n$ point sites with respect to a set $S$ of $m$ points. A site $p \in P$ is non-dominated if and only if for each $q \in P \setminus \{p\}$, there exists at least one point $s \in S$ that is closer to $p$ than to $q$. We reduce this problem of determining non-dominated sites to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under a convex distance function. The weights of said Voronoi diagram are derived from the coordinates of the sites of $P$, while the convex distance function is derived from $S$. In the two-dimensional plane, this reduction gives an $O((n + m) \log (n + m))$-time algorithm to find the non-dominated points.

cs.CG↗

arXiv2018

Minsum $k$-Sink Problem on Path Networks

We consider the problem of locating a set of $k$ sinks on a path network with general edge capacities that minimizes the sum of the evacuation times of all evacuees. We first present an $O(kn\log^4n)$ time algorithm when the edge capacities are non-uniform, where $n$ is the number of vertices. We then present an $O(kn\log^3 n)$ time algorithm when the edge capacities are uniform. We also present an $O(n\log n)$ time algorithm for the special case where $k=1$ and the edge capacities are non-uniform.

cs.DS↗

arXiv2018

Minmax Regret 1-Sink for Aggregate Evacuation Time on Path Networks

Evacuation in emergency situations can be modeled by a dynamic flow network. Two criteria have been used before: one is the evacuation completion time and the other is the aggregate evacuation time of individual evacuees. The aim of this paper is to optimize the aggregate evacuation time in the simplest case, where the network is a path and only one evacuation center (called a sink) is to be introduced. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute the sink location that minimizes the maximum "regret." We present an $O(n^2\log n)$ time algorithm to solve this problem, improving upon the previously fastest $O(n^3)$ time algorithm, where $n$ is the number of vertices.

cs.DS↗

arXiv2017

Bilinear Assignment Problem: Large Neighborhoods and Experimental Analysis of Algorithms

The bilinear assignment problem (BAP) is a generalization of the well-known quadratic assignment problem (QAP). In this paper, we study the problem from the computational analysis point of view. Several classes of neigborhood structures are introduced for the problem along with some theoretical analysis. These neighborhoods are then explored within a local search and a variable neighborhood search frameworks with multistart to generate robust heuristic algorithms. Results of systematic experimental analysis have been presented which divulge the effectiveness of our algorithms. In addition, we present several very fast construction heuristics. Our experimental results disclosed some interesting properties of the BAP model, different from those of comparable models. This is the first thorough experimental analysis of algorithms on BAP. We have also introduced benchmark test instances that can be used for future experiments on exact and heuristic algorithms for the problem.

cs.DS↗

arXiv2016

Improved Algorithms for Computing $k$-Sink on Dynamic Path Networks

We present a novel approach to finding the $k$-sink on dynamic path networks with general edge capacities. Our first algorithm runs in $O(n \log n + k^2 \log^4 n)$ time, where $n$ is the number of vertices on the given path, and our second algorithm runs in $O(n \log^3 n)$ time. Together, they improve upon the previously most efficient $O(kn \log^2 n)$ time algorithm due to Arumugam et al. for all values of $k$. In the case where all the edges have the same capacity, we again present two algorithms that run in $O(n + k^2 \log^2n)$ time and $O(n \log n)$ time, respectively, and they together improve upon the previously best $O(kn)$ time algorithm due to Higashikawa et al. for all values of $k$.

cs.DS↗

arXiv2016

Linear-Time Fitting of a $k$-Step Function

Given a set of $n$ weighted points on the $x$-$y$ plane, we want to find a step function consisting of $k$ horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in $O(n)$ time when $k$ is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-constrained k-center problem and the size-$k$ histogram construction problem as well.

cs.CG↗

arXiv2016

The Bilinear Assignment Problem: Complexity and polynomially solvable special cases

In this paper we study the {\it bilinear assignment problem} (BAP) with size parameters $m$ and $n$, $m\leq n$. BAP is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence NP-hard. We show that BAP cannot be approximated within a constant factor unless P=NP even if the associated quadratic cost matrix $Q$ is diagonal. Further, we show that BAP remains NP-hard if $m = O(\sqrt[r]{n})$, for some fixed $r$, but is solvable in polynomial time if $m = O(\sqrt{\log n})$. When the rank of $Q$ is fixed, BAP is observed to admit FPTAS and when this rank is one, it is solvable in polynomial time under some additional restrictions. We then provide a necessary and sufficient condition for BAP to be equivalent to two linear assignment problems. A closed form expression to compute the average of the objective function values of all solutions is presented, whereas the median of the solution values cannot be identified in polynomial time, unless P=NP. We then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of $(m-1)!(n-1)!$ solutions. However, computing a solution whose objective function value is no worse than that of $m!n!-\lceil\frac{m}β\rceil !\lceil\frac{n}β\rceil !$ solutions is NP-hard for any fixed rational number $β>1$.

math.OC↗

arXiv2016

Geometric k-Center Problems with Centers Constrained to Two Lines

We consider the $k$-center problem in which the centers are constrained to lie on two lines. Given a set of $n$ weighted points in the plane, we want to locate up to $k$ centers on two parallel lines. We present an $O(n\log^2 n)$ time algorithm, which minimizes the weighted distance from any point to a center. We then consider the unweighted case, where the centers are constrained to be on two perpendicular lines. Our algorithms run in $O(n\log^2 n)$ time also in this case.

cs.CG↗

arXiv2016

The $p$-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983].

cs.DS↗

arXiv2015

Approximation Algorithms for Generalized MST and TSP in Grid Clusters

We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each grid cell is $1 \times 1$. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a $(1+4\sqrt{2}+ε)$ and $(1.5+8\sqrt{2}+ε)$-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further.

cs.DM↗