Aritra Banik

Choice Is Hard

Let \(P=\{C_1,C_2,\ldots , C_n\}\) be a set of color classes, where each color class \(C_i\) consists of a pair of objects. We focus on two problems in which the objects are points on the line. In the first problem (rainbow minmax gap), given P, one needs to select exactly one point from each color class, such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan, who left the question of its complexity unresolved. We prove that it is NP-hard. For our proof we obtain the following auxiliary result. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We prove that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. We present two additional applications of the LSAT result, namely, to rainbow piercing and rainbow covering.

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 .

The Discrete Voronoi Game in a Simple Polygon

Let P be a simple polygon with m vertices and let \(\mathcal{U}\) be a set of n points in P. We consider the points of \(\mathcal{U}\) to be “users”. We consider a game with two players \(\mathcal{P}_1\) and \(\mathcal{P}_2\). In this game, \(\mathcal{P}_1\) places a point facility inside P, after which \(\mathcal{P}_2\) places another point facility inside P. We say that a user \(u \in \mathcal{U}\) is served by its nearest facility, where distances are measured by the geodesic distance in P. The objective of each player is to maximize the number of users they serve. We show that for any given placement of a facility by \(\mathcal{P}_1\), an optimal placement for \(\mathcal{P}_2\) can be computed in O(m + n(logn + logm)) time. We also provide a polynomial-time algorithm for computing an optimal placement for \(\mathcal{P}_1\).

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.

The p-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete

Discrete Voronoi games and ϵ-nets, in two and three dimensions

p

Minimum enclosing circle of a set of fixed points and a mobile point

-center problem for tree networks with

A Polynomial Sized Kernel for Tracking Paths Problem.

n

Some (in)tractable Parameterizations of Coloring and List-Coloring

vertices. One of our proposed algorithms runs in

Selecting and covering colored points

O(n \log n + p \log^2 n \log(n/p))