A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions
reducible to a finite number of polynomial equalities and inequalities, it is shown how to …

From the winner of the Turing Award and the Abel Prize, an introduction to computational
complexity theory, its connections and interactions with mathematics, and its central role in …

We introduce a method for proving lower bounds on the efficacy of semidefinite
programming (SDP) relaxations for combinatorial problems. In particular, we show that the …

Over the last twenty years, an exciting interplay has emerged between proof systems and
algorithms. Some natural families of algorithms can be viewed as a generic translation from …

This chapter gives an overview of proof complexity and connections to SAT solving, focusing
on proof systems such as resolution, Nullstellensatz, polynomial calculus, and cutting planes …

In this paper we establish a linear (thereby, sharp) lower bound on degrees of
Positivstellensatz calculus refutations over a real field introduced in Grigoriev and Vorobjov …

In order to obtain the best-known guarantees, algorithms are traditionally tailored to the
particular problem we want to solve. Two recent developments, the Unique Games …

Finding cliques in random graphs and the closely related" planted" clique variant, where a
clique of size k is planted in a random G (n, 1/2) graph, have been the focus of substantial …

Given a random $ n\times n $ symmetric matrix $\boldsymbol W $ drawn from the Gaussian
orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the …

We present a general approach to rounding semidefinite programming relaxations obtained
by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the …