# Voronoi Game on Graphs.

December 2013

## Date of Award

Winter 12-12-2014

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Advance Computing and Microelectronics Unit (ACMU-Kolkata)

## Supervisor

Das, Sandip (ACMU-Kolkata; ISI)

## Abstract (Summary of the Work)

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 thesis we consider one round Voronoi game with two players. Here the underlying space is a road network, which is modeled by a graph embedded in R2. In this game each of the players places a set of facilities and the underlying graph is subdivided according to the nearest neighbour rule. The player which dominates the maximum region of the graph wins the game. This thesis mainly deals with the problem of determining optimal strategies of the players. We characterize the optimal facility locations of second player given a placement by first player. Using this result we design a polytime algorithm for determining the optimal strategy of second player on trees. We also show that the same problem is P-hard when the underlying space is a general graph. Moreover we present a 1.58 factor approximation algorithm for the above mentioned problem. Then we concentrate on optimal strategy of first player. We give a lower bound on the optimal payoff of first player. We discuss optimal strategy of first player for (1, 1) and (2, 1) game on tree network. Then we characterize optimal facility locations of first player for (1, 1) game on graph network.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843202

## Control Number

ISI-DISS-2013-302