**Speaker: Tselil Schramm**, University of California, Berkeley

**Title**: Strongly refuting random constraint satisfaction problems below the spectral threshold

**Abstract: **

Random instances of 3SAT are known to be unsatisfiable with high probability when there at least 5N

clauses. However, given a random 3SAT instance on N variables, the task of refuting the instance, or of certifying that the instance has no satisfying assignments, is hypothesized to be computationally hard if there are O(N) clauses—in fact, the best known efficient algorithms for refutation require instances with at least N3/2 clauses, a factor of N1/2 beyond the unsatisfiability threshold.

In this talk, I will describe a new spectral algorithm that refutes 3SAT instances with fewer clauses, given more time. Our algorithm extends the best polynomial time algorithms at N3/2 clauses, interpolating between them and the exponential brute-force search at the unsatisfiability threshold at Θ(N) clauses. Further, our algorithm strongly refutes the instances, certifying that no assignment satisfies more than a (7/8)-fraction of constraints.

**Biography:**

Tselil Schramm is a fifth year PhD student in Computer Science at UC Berkeley, advised by Prasad Raghavendra and Satish Rao, and supported by a National Science Foundation Fellowship and a UC Berkeley Chancellor’s Fellowship. Her research interests include spectral algorithms, spectral graph theory, convex programming, and approximation algorithms.