Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

On the pseudo-deterministic query complexity of NP search problems

We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse

Randomized communication versus partition number

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique versus Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Göös, Pitassi, and Watson, FOCS\u2715). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs

Randomized communication vs. partition number

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique vs. Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Göös, Pitassi, and Watson, FOCS 2015). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs

Reducing the complexity of reductions

We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a "Gap Theorem", showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper: ¶1. Does the "gap" between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0 reductions? ¶2. Does a uniform version of the isomorphism theorem hold? ¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits? ¶ We answer all of these questions. In particular, we prove that the Berman-Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class closed under uniform TC0-computable many-one reductions, the following three theorems hold: ¶1. If contains sets that are complete under a notion of reduction at least as strong as Dlogtime-uniform AC0[mod 2] reductions, then there are such sets that are not complete under (even non-uniform) AC0 reductions. ¶2. The sets complete for under P-uniform AC0 reductions are all isomorphic under isomorphisms computable and invertible by P-uniform AC0 circuits of depth-three. ¶3. There are sets complete for under Dlogtime-uniform AC0 reductions that are not isomorphic under any isomorphism computed by (even non-uniform) AC0 circuits of depth two. ¶To prove the second theorem, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.