Separation of the Factorization Norm and Randomized Communication Complexity - Cheung, Tsun-Ming; Hatami, Hamed; Hosseini, Kaave; Shirley, Morgan (Computational complexity, 2025)
In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ 2 -factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ 2 norm with the γ 2 norm holds. We answer the question of Linial and Shraibman in the negative by exhibiting a 2 n × 2 n Boolean matrix with γ 2 norm 2 Ω ( n ) and randomized communication complexity O ( log n ) . As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITCS '23). Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ 2 norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.