# Fine-Grained Rough Draft

Aaron Potechin (Massachusetts Institute of Technology) and Ivan Mikhailin (UC San Diego)

Calvin Lab Room 116

## Speaker: Aaron Potechin

## Sum of Squares Lower Bounds for the Planted Clique Problem

The planted clique problem asks whether or not we can find a k-clique which is planted in a random G(n,1/2) graph. Although the expected size of the largest clique in a random graph is only 2(lg(n)), we currently do not know of any polynomial time algorithm which can find a planted clique of size much smaller than n^(1/2).The sum of squares hierarchy gives a hierarchy of approximation algorithms, each more powerful than the last. These algorithms are some of the most powerful approximation algorithms we know of, but their performance on many problems is not well understood. In fact, until very recently there were no non-trivial bounds on the performance of the sum of squares hierarchy on the planted clique problem.In this talk, I will present the first non-trivial bounds for this question, describing the planted clique problem, the sum of squares hierarchy, and how we can show that the sum of squares hierarchy fails to find the planted clique if its size k is sufficiently small.

## Speaker: Ivan Mikhailin

Solving Graph Problems Using FFT

Despite many decades of research for some of the classical graph problems like Traveling salesman problem or Coloring problem no algorithm with running time less then 2^n is known. This bound looks like a natural barrier and improving over it would be a groundbreaking result. However for case of sparse graphs, some recent progress has been made. In the talk I will describe a unified technique based on FFT to obtain exponential speed up for a problems on a sparse graphs.