Thesis Oral Defense - Peter Manohar

— 1:00pm

In Person and Virtual - ET - Reddy Conference Room, Gates Hillman 4405 and Zoom

PETER MANOHAR , Ph.D. Candidate, Computer Science Department, Carnegie Mellon University

New Spectral Techniques in Algorithms, Combinatorics, and Coding Theory: The Kikuchi Matrix Method

In this thesis, we present a new method to solve algorithmic and combinatorial problems by (1) reducing them to bounding the maximum, over x in {-1,1}n, of homogeneous degree-q multilinear polynomials, and then (2) bounding the maximum value attained by these polynomials by analyzing the spectral properties of appropriately chosen induced subgraphs of Cayley graphs on the hypercube (and related variants) called "Kikuchi matrices". We will present the following applications of this method.

  1. Designing algorithms for refuting/solving semirandom and smoothed instances of constraint satisfaction problems;
  2. Proving Feige's conjectured hypergraph Moore bound on the extremal girth vs. density trade-off for hypergraphs;
  3. Proving a cubic lower bound for 3-query locally decodable codes and an exponential lower bound for 3-query locally correctable codes.

Thesis Committee: 

Venkatesan Guruswami (Co-Chair, Carnegie Mellon University / University of California, Berkeley)
Pravesh K. Kothari (Co-Chair, Carnegie Mellon University / Princeton University)
Ryan O’Donnell
Uriel Feige (Weizmann Institute)

In Person and Zoom Participation.  See announcement.

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