Approximation Algorithms for Directed Steiner Tree and Traveling Salesman Problems

• Author / Creator

  Mousavi Haji, Seyyed Ramin

• In this thesis, we design approximation algorithms for a variety of problems in Network Design. The first problem we consider is the Directed Steiner Tree (DST) problem where we want to find a cheapest way of connecting a subset of nodes (terminal nodes) from a root node in a directed network. We give the first logarithmic approximation algorithm for DST on planar graphs.

  Another restriction of DST we consider is to quasi-bipartite instances on planar graphs. In these instances the underlying graph is planar and no two non-terminal nodes are adjacent. Here we get the first constant factor approximation. We further extend this result to a more general family of graphs called minor-free graphs. Our approach is based on a non-standard primal-dual framework and also bounds the integrality gap of the classical linear programming (LP) relaxation for DST.

  The third problem we study is a variant of the Traveling Salesman Problem (TSP), one the most famous problem in combinatorial optimization. In TSP, we are given a set of cities, the task is to find a minimum cost tour (closed walk) visiting all the cities. We initiate the study of generalization of TSP, where we are interested in tours that respect some given degree bounds, i.e., a feasible tour must not pass through a location more than a given bound. We further generalize this problem to the setting where we only need to visit a subset of locations in the network. It is easy to see unless P=NP, we cannot hope for an algorithm that does not violate the degree bounds at all. On the other hand, the problems we consider are a generalization of TSP, therefore approximating the cost factor better than the approximation factor for TSP is a very challenging problem. In this thesis, we study the trade-offs between the cost of the tour and the degree violation of the nodes in the tour. We develop LP-based rounding algorithms for these TSP variants which in turn bound the integrality gap of a natural LP relaxations for these problems.

• Subjects / Keywords

  • Approximation algorithms
  • Directed Steiner tree
  • Traveling salesman problem

• Graduation date

  Fall 2023

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-tm9a-9c91

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.