Communication Complexity of Cake Cutting

We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.Comment: Added efficient communication protocol for the monotone crossing proble

Strengths and Weaknesses of Quantum Fingerprinting

We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical shared-randomness SMP protocols by means of quantum SMP protocols without shared randomness ($Q^\parallel$-protocols). Our first result is to extend Yao's simulation to the strongest possible model: every many-round quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by $Q^\parallel$-protocols. We apply our technique to obtain an efficient $Q^\parallel$-protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.Comment: 13 pages, no figures, to appear in CCC'0

New bounds on classical and quantum one-way communication complexity

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical model, our bound extends the well known upper bound of Kremer, Nisan and Ron to include non-product distributions. We show that for a boolean function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where D_{epsilon}^{1, mu}(f) represents the one-way distributional communication complexity of f with error at most epsilon under mu; vc(f) represents the Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual information, under mu, between the random inputs of the two parties. For a non-boolean function f:X x Y ->[k], we show a similar upper bound on D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' = f/k. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f, in terms the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f . We show for a non-boolean total function f : X x Y -> Z and a product distribution mu on XxY, Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional communication complexity of f with error at most epsilon^3/8 under mu and rec_ epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x 15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).Comment: ver 1, 19 page