Abstract
We study two basic graph parameters, the chromatic number and the orthogonal rank, in the context of classical and quantum exact communication complexity. In particular, we consider two types of communication problems that we call promise equality and list problems. For both of these, it was already known that the oneround classical and oneround quantum complexities are characterized by the chromatic number and orthogonal rank of a certain graph, respectively.
In a promise equality problem, Alice and Bob must decide if their inputs are equal or not. We prove that classical protocols for such problems can always be reduced to oneround protocols with no extra communication. In contrast, we give an explicit instance of a promise problem that exhibits an exponential gap between the one and tworound exact quantum communication complexities. Whereas the chromatic number thus captures the complete complexity of promise equality problems, the hierarchy of "quantum chromatic numbers" (starting with the orthogonal rank) giving the quantum communication complexity for every fixed number of communication rounds thus turns out to enjoy a much richer structure.
In a list problem, Bob gets a subset of some finite universe, Alice gets an element from Bob's subset, and their goal is for Bob to learn which element Alice was given. The best general lower bound (due to Orlitsky) and upper bound (due to Naor, Orlitsky, and Shor) on the classical communication complexity of such problems differ only by a constant factor. We exhibit an example showing that, somewhat surprisingly, the fourround protocol used in the bound of Naor et al. can in fact be optimal. Finally, we pose a conjecture on the orthogonality rank of a certain graph whose truth would imply an intriguing impossibility of round elimination in quantum protocols for list problems, something that works trivially in the classical case.
BibTeX  Entry
@InProceedings{brit_et_al:LIPIcs:2015:5558,
author = {Jop Bri{\"e}t and Harry Buhrman and Debbie Leung and Teresa Piovesan and Florian Speelman},
title = {{Round Elimination in Exact Communication Complexity}},
booktitle = {10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
pages = {206225},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897965},
ISSN = {18688969},
year = {2015},
volume = {44},
editor = {Salman Beigi and Robert Koenig},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5558},
URN = {urn:nbn:de:0030drops55588},
doi = {10.4230/LIPIcs.TQC.2015.206},
annote = {Keywords: communication complexity, round elimination, quantum communication, protocols, chromatic numbers}
}
Keywords: 

communication complexity, round elimination, quantum communication, protocols, chromatic numbers 
Collection: 

10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015) 
Issue Date: 

2015 
Date of publication: 

04.11.2015 