The main focus of this thesis work is computational aspects of discrepancy theory. Discrepancy theory studies how well discrete objects can approximate continuous ones. This question is ubiquitous in mathematics and computer science, and discrepancy theory has found numerous applications. In this thesis work, we (1) initiate the study of the polynomial time approximability of central discrepancy measures: we prove the first hardness of approximation results and design the first polynomial time approximation algorithms for combinatorial and hereditary discrepancy. We also (2) make progress on longstanding open problems in discrepancy theory, using insights from computer science: we give nearly tight hereditary discrepancy lower bounds for axis-aligned boxes in higher dimensions, and for homogeneous arithmetic progressions. Finally, we have (3) found new applications of discrepancy theory to (3a) fundamental questions in private data analysis and to (3b) communication complexity. In particular, we use discrepancy theory to design nearly optimal efficient algorithms for counting queries, in all parameter regimes considered in the literature. We also show that discrepancy lower bounds imply communication lower bounds for approximation problems in the one-way model. Directions for further research and connections to expander graphs, compressed sensing, and the design of approximation algorithms are outlined.
Subject (authority = RUETD)
Topic
Computer Science
Subject (authority = ETD-LCSH)
Topic
Irregularities of distribution (Number theory)
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_5939
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (ix, 172 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Aleksandar Nikolov
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
Rutgers University. Graduate School - New Brunswick
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Type
License
Name
Author Agreement License
Detail
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