Approximation algorithms for submodular optimization and graph problems

In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NP-hard and therefore, assuming that P =/= NP, there do not exist polynomial-time 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 polynomial-time 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 non-metric 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 poly-logarithmic approximation with constant congestion for the node-disjoint paths problem and we show a poly-logarithmic upper bound on the gap between the maximum fractional and integral throughput flows in node-capacitated 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 poly-logarithmic approximation with constant congestion for the all-or-nothing 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 prize-collecting 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 minor-free graphs --- leads to algorithms whose approximation guarantees are a significant improvement over what can be achieved for general graphs.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Ene, Alina

Contributors dc:contributor

• Chekuri, Chandra S.
• Godfrey, Philip B.
• Har-Peled, Sariel
• Vondrak, Jan

• Approximation algorithms
• Submodular optimization
• Routing
• Network design

Statement dc:rights

• Copyright 2013 Alina Ene

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/46738

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/46738