Hardness and pproximation of some Graph Theoretic problems

Abstract

In the real world, we encounter many problems that can be modeled as graphtheoretic problems. This modeling gives a concrete view of the constraints and objectives of the problem and allows us to apply some well-known techniques to solve it. Many of these problems do not have their computational complexity settled; on the other hand, many others have been proved to be NP-hard. Thus should be approximated. This thesis focuses on these aspects of some graph theoretic problems. The Traveling Tournament Problem is one of the interests of this thesis. A constrained Traveling Tournament Problem(TTP-k) asks for a schedule of a double round-robin tournament with an upper bound(k) on the lengths of home stands and away trips of the teams where the total travel distance is minimized. The hardness of the problem varies with the upper bound. This thesis attempts an approximation algorithm for TTP-2, which is assumed to be NP-Hard. Then considers a study on the hardness analysis of TTP-k where k > 3 and k ∈ N. The Firefighter Problem is an important graph theoretic problem with practical application in a recent pandemic scenario. The firefighter problem asks for a solution to save vertices in a graph by placing firefighters on some of them where a fire broke out in a vertex and spread through the network with time. This thesis considers Firefighter Problem on Unit Disk Graphs. Most networks can be modeled in this wireless era as Unit Disk Graphs. The hardness of the problem and an approximation algorithm for the same is attempted in this thesis. Then a special version of the firefighter problem called the Firebreak Problem is considered where the firefighters can be placed on the vertices only at the initial time instance when the fire breaks out. An approximation algorithm is attempted for the Firebreak Problem on Split Graphs which has been proven to be NP-Hard.