Multiparty Communication Complexity of Collision Finding

TL;DR

It is proved an $\Omega(n-1-1/k)$ lower bound on the k-party number-in-hand communication complexity of collision-finding and a lower bound on the size of tree-like cutting-planes proofs of the bit pigeonhole principle is proved.

Abstract

We prove an $Ω(n^{1-1/k} \log k \ /2^k)$ lower bound on the $k$-party number-in-hand communication complexity of collision-finding. This implies a $2^{n^{1-o(1)}}$ lower bound on the size of tree-like cutting-planes proofs of the bit pigeonhole principle, a compact and natural propositional encoding of the pigeonhole principle, improving on the best previous lower bound of $2^{Ω(\sqrt{n})}$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Threshold Decision Tree
• Proposition 2.2
• Theorem 2.3: braverman-oshman-nih-disj
• Lemma 2.4
• Proposition 2.5: muroga-bookmuroga1961theory
• Proposition 2.6: viola2015communication-addition
• Corollary 2.7
• proof : Proof of \ref{['lem:short-protocol-from-thr-dt']}