NEW COMPUTATIONAL ASPECTS OF DISCREPANCY THEORY

Abstract

OF THE DISSERTATION New Computational Aspects of Discrepancy Theory by Aleksandar Nikolov Dissertation Director: S. Muthukrishnan 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.