Sayan Bandyapadhyay

Voronoi game on graphs

Voronoi game is a geometric model of competitive facility location problem played between two players. Users are generally modeled as points uniformly distributed on a given underlying space. Each player chooses a set of points in the underlying space to place their facilities. Each user avails service from its nearest facility. Service zone of a facility consists of the set of users which are closer to it than any other facility. Payoff of each player is defined by the quantity of users served by all of its facilities. The objective of each player is to maximize their respective payoff. In this paper we consider the two player Voronoi game where the underlying space is a road network modeled by a graph. In this framework we consider the problem of finding k optimal facility locations of Player 2 given any placement of m facilities by Player 1. Our main result is a dynamic programming based polynomial time algorithm for this problem on tree network. On the other hand, we show that the problem is strongly NP -complete for graphs. This proves that finding a winning strategy of P2 is NP -complete. Consequently, we design a 1 - 1 e factor approximation algorithm, where e ? 2.718 .

Voronoi Game on Graphs

Voronoi game is a geometric model of competitive facility location problem, where each market player comes up with a set of possible locations for placing their facilities. The objective of each player is to maximize the region occupied on the underlying space. In this paper we consider one round Voronoi game with two players. Here the underlying space is a road network, which is modeled by a graph embedded on ℝ2. In this game each of the players places a set of facilities and the underlying graph is subdivided according to the nearest neighbor rule. The player which dominates the maximum region of the graph wins. Given a placement of facilities by Player 1, we have characterized the optimal placement by Player 2. At first we dealt with the case when Player 2 places a constant number of facilities and provided an algorithm for the same. Next we have proved that finding the optimal placement of k facilities by Player 2 is \(\mathcal{NP}\)-hard where k is given. Lastly we presented a 1.58 factor approximation algorithm for the above mentioned problem.

Near-Optimal Clustering in the k-machine model

The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The k-machine model (Klauck et al., SODA 2015) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the p-median problem, and the p-center problem and presents O (1)-factor approximation algorithms for these problems running in Õ (n/k) rounds in the k -machine model. These algorithms are optimal upto polylogarithmic factors because this paper also shows Ω (n/k) lower bounds for obtaining poly(n)-factor approximation algorithms for these problems. These are the first results for clustering problems in the k -machine model. We assume that the metric provided as input for these clustering problems in only implicitly provided, as an edge-weighted graph and in a nutshell, our main technical contribution is to show that constant-factor approximation algorithms for all three clustering problems can be obtained by learning only a small portion of the input metric.

Effectiveness of Local Search for Art Gallery Problems

We study the variant of the art gallery problem where we are given an orthogonal polygon P (possibly with holes) and we want to guard it with the minimum number of sliding cameras. A sliding camera travels back and forth along an orthogonal line segment s in P and a point p in P is said to be visible to the segment s if the perpendicular from p onto s lies in P. Our objective is to compute a set containing the minimum number of sliding cameras (orthogonal segments) such that every point in P is visible to some sliding camera. We study the following two variants of this problem: Minimum Sliding Cameras problem, where the cameras can slide along either horizontal or vertical segments in P, and Minimum Horizontal Sliding Cameras problem, where the cameras are restricted to slide along horizontal segments only. In this work, we design local search PTASes for these two problems improving over the existing constant factor approximation algorithms. We note that in the first problem, the polygons are not allowed to contain holes. In fact, there is a family of polygons with holes for which the performance of our local search algorithm is arbitrarily bad.

Approximate Clustering via Metric Partitioning

In this paper we consider two metric covering/clustering problems - \textit{Minimum Cost Covering Problem} (MCC) and

Polynomial Time Algorithms for Bichromatic Problems

k

On the Approximability of Orthogonal Order Preserving Layout Adjustment

-clustering. In the MCC problem, we are given two point sets

On Variants of k-means Clustering.

X

Approximation schemes for partitioning: convex decomposition and surface approximation

(clients) and

Capacitated Covering Problems in Geometric Spaces

Y