Files in this item
Files  Description  Format 

application/pdf Alina_Ene.pdf (1MB)  (no description provided) 
Description
Title:  Approximation algorithms for submodular optimization and graph problems 
Author(s):  Ene, Alina 
Director of Research:  Chekuri, Chandra S. 
Doctoral Committee Chair(s):  Chekuri, Chandra S. 
Doctoral Committee Member(s):  Godfrey, Philip B.; HarPeled, Sariel; Vondrak, Jan 
Department / Program:  Computer Science 
Discipline:  Computer Science 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Approximation algorithms
Submodular optimization Routing Network design 
Abstract:  In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P =/= NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intractability of these problems, we focus on algorithms that construct approximate solutions: An approximation algorithm is a polynomialtime algorithm that, for any instance of the problem, it outputs a solution whose value is within a multiplicative factor p of the value of the optimal solution for the instance. The quantity p is the approximation ratio of the algorithm and we aim to achieve the smallest ratio possible. Our focus in this thesis is on designing approximation algorithms for several combinatorial optimization problems. In the first part of this thesis, we study a class of constrained submodular minimization problems. We introduce a model that captures allocation problems with submodular costs and we give a generic approach for designing approximation algorithms for problems in this model. Our model captures several problems of interest, such as nonmetric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization to hub location. Using a convex relaxation and rounding strategy, we achieve good approximation guarantees for several problems in this model. In particular, we match or improve the known approximation ratios for several problems in a unified fashion. In the second part of this thesis, we study muticommodity flow problems in both undirected and directed graphs and we make several contributions towards understanding the gap between fractional and integral multicommodity flows. We give a polylogarithmic approximation with constant congestion for the nodedisjoint paths problem and we show a polylogarithmic upper bound on the gap between the maximum fractional and integral throughput flows in nodecapacitated undirected graphs. Prior to our work, the best guarantees were only polynomial. In the process, we prove a conjecture of Chekuri, Khanna, and Shepherd on the connection between the treewidth of the graph and the existence of a good routing structure. Additionally, we initiate the study of integral throughput flow problems in directed graphs with symmetric demand pairs. We obtain a polylogarithmic approximation with constant congestion for the allornothing flow problem. In the third part of this thesis, we study several network design problems. The input to these problems is a graph with costs on the edges or the nodes and the output is a minimum cost subgraph that meets certain connectivity requirements. We study several network design problems in planar graphs, including the prizecollecting Steiner tree and forest and the survivable network design problem with node costs. We show that the special structure of planar graphs  and more generally, bounded genus and minorfree graphs  leads to algorithms whose approximation guarantees are a significant improvement over what can be achieved for general graphs. 
Issue Date:  20140116 
URI:  http://hdl.handle.net/2142/46738 
Rights Information:  Copyright 2013 Alina Ene 
Date Available in IDEALS:  20140116 
Date Deposited:  201312 
This item appears in the following Collection(s)

Dissertations and Theses  Computer Science
Dissertations and Theses from the Dept. of Computer Science 
Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois