We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph $H$, the $H$-colouring problem is to decide whether a given graph has a homomorphism to $H$. By a result of Hell and Nešet?il, this problem is NP-hard for any non-bipartite graph $H$. In ... more >>>

We show that if a $k$-CNF requires width $w$ to refute in resolution, then it requires space $\sqrt w$ to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is ... more >>>

We exhibit a pseudorandom generator with nearly quadratic stretch for randomized Turing machines, which have a one-way random tape and a two-way work tape. This is the first generator for this model. Its stretch is essentially the best possible given current lower bounds. We use the generator to prove a ... more >>>