Complexity of Permutation-Invariant Functions

Motivated by the quest for a broader understanding of communication complexity of simple functions, we introduce the class of “permutation-invariant” functions. A partial function f : {0,1} × {0,1} → {0, 1, ?} is permutation-invariant if for every bijection π : {1, . . . , n} → {1, . . . , n} and every x,y ∈ {0, 1}, it is the case that f (x,y) = f (x,y). Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in n given an implicit description of f ) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as SET-DISJOINTNESS and INDEXING, while complementing them with the relatively lesser-known upper bounds for GAP-INNER-PRODUCT (from the sketching literature) and SPARSE-GAP-INNER-PRODUCT (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive O(log log n) overhead. ∗Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge MA 02139. Supported in part by NSF STC Award CCF 0939370 and NSF Award CCF-1217423. badih@mit.edu. †Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge MA 02139. Supported by NSF CCF-1420956. pritish@mit.edu. ‡Microsoft Research, One Memorial Drive, Cambridge, MA 02142, USA. madhu@mit.edu. 1 ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 87 (2015)