Vinyals 1 Recap of Last Lecture and Plans for Today

on resolution refutations on pigeonhole principle formulas PHP n . Observe that PHP n has size N = Θ(n3), so the lower bound is, in fact, exp ( Ω ( 3 √ N )) in terms of formula size, and not truly exponential (by which we mean exp(Ω(N)) with the coefficient in the exponent scaling linearly with N ). In order to prove this lower bound we used the Prosecutor–Defendant game, where Prosecutor asks whether pigeon i flies into hole j, and Defendant replies in a way that delays an explicit contradiction for as long as possible. Good Defendant strategies for this game imply resolution lower bounds, so we constructed such a strategy: Defendant picks a random matching of n/4 pigeons to n/4 pigeonholes, and then answers according to this matching when asked about pigeons in this matching, and otherwise for other pigeons says that they do not go into holes for as long as possible. It is easy to see that there are exponentially many different choices for a matching. It takes some more work to show that, before Prosecutor wins, there has be be a record where a noticeable fraction of (information about) this random matching is written down. Once this is shown, however, it follows that Prosecutor needs exponentially many records, which immediately gives an exponential resolution lower bound. Today we look at formulas encoding (a contradiction of) the handshaking lemma: “The sum of vertex degrees in an undirected graphG = (V,E) is even” or, in math symbols, that ∑ v∈V deg(v) ≡ 0 (mod 2). Our goal is to obtain lower bounds that are truly exponential in the size of the formula.