Separation of the Factorization Norm and Randomized Communication Complexity

• Published in: Computational complexity Vol. 34; no. 2; p. 17
• Main Authors: Cheung, Tsun-Ming, Hatami, Hamed, Hosseini, Kaave, Shirley, Morgan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Communication; Complexity; Computational Mathematics and Numerical Analysis; Computer Science; Factorization; Lower bounds; Questions; Separation; 68Q11; Factorization norms; randomized communication complexity
• ISSN: 1016-3328; 1420-8954; 1420-8954
• DOI: 10.1007/s00037-025-00278-3

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.