Raghu Meka, Avi Wigderson

Finding cliques in random graphs and the closely related ``planted'' clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = ... more >>>

Christoph Berkholz

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and sum-of-squares (a.k.a. Lasserre), which are based on linear and semi-definite programming relaxations. On the other hand, we consider polynomial calculus, ... more >>>

Noah Fleming, Pravesh Kothari, Toniann Pitassi

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 a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential ... more >>>

Yaroslav Alekseev, Dima Grigoriev, Edward Hirsch, Iddo Tzameret

We introduce the `binary value principle' which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, ... more >>>

Shuo Pang

We prove a SOS degree lower bound for the planted clique problem on Erd{\"o}s-R\'enyi random graphs $G(n,1/2)$. The bound we get is degree $d=\Omega(\epsilon^2\log n/\log\log n)$ for clique size $\omega=n^{1/2-\epsilon}$, which is almost tight. This improves the result of \cite{barak2019nearly} on the ``soft'' version of the problem, where the family ... more >>>