Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations. We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for t...

The symmetric traveling salesman problem (STSP) is a special case of ATSP in which the cost function is symmetric (i.e. c(u, v) = c(v, u)). Christofides in 1976 [6] discovered a 3/2approximation algorithm for STSP. No better approximation algorithm has since been found for the general symmetric metric case. However, polynomial-time approximation schemes have been found when the distances are sh...

The paper provides a description of the two recent approximation algorithms for the Asymmetric Traveling Salesman Problem, giving the intuitive description of the works of Feige-Singh[1] and Asadpour et.al [2]. [1] improves the previous O(log n) approximation algorithm, by improving the constant from 0.84 to 0.66 and modifying the work of Kaplan et. al [3] and also shows an efficient reduction ...

In the next few lectures we will talk about three new variants of the classical randomized rounding technique, namely the rounding by sampling method, the pipage rounding method and the iterative rounding method. We will use these techniques to design approximation algorithms for Asymmetric TSP (ATSP) and the Steiner tree problems. The materials of the this lecture are based on the works of Asa...

The symmetric traveling salesman problem (STSP) is a special case of ATSP in which the cost function is symmetric (i.e. c(u, v) = c(v, u)). Christofides in 1976 [6] discovered a 3/2approximation algorithm for STSP. No better approximation algorithm has since been found for the general symmetric metric case. However, polynomial-time approximation schemes have been found when the distances are sh...

A polyhedral study of the asymmetric travelling salesman problem with time windows Abstract The asymmetric travelling salesman problem with time windows (ATSP-TW) is a basic model for scheduling and routing applications. In this paper we present a formulation of the problem involving only 0/1-variables associated with the arcs of the underlying digraph. This has the advantage of avoiding additi...