This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, one-way communication, and applications of quantum communication complexity. 1

this article is on protocols allowing the well-functioning parts of such a large and complex system to carry out their work despite the failure of others. Many deep and interesting results on such problems have been discovered by computer scientists in recent years, the incorporation of which into game theory can greatly enrich this field

We study the following problem: two agents Alice and Bob are connected to each other by independent discrete memoryless channels. They wish to generate common randomness, i.e., agree on a common random variable, by communicating interactively over the two channels. Assuming that Alice and Bob are allowed access to independent external random sources at rates (in bits per step of communication) of HA and HB , respectively, we show that they can generate common randomness at a rate of maxfmin[HA + H(W j Q);I(P;V)] + min[HB + H(V j P );I(Q;W)]g bits per step, by exploiting the noise on the two channels. Here, V is the channel from Alice to Bob, and W is the channel from Bob to Alice. The maximum is over all probability distributions P and Q on the input alphabets of V and W , respectively. We also prove a strong converse which establishes the above rate as the highest attainable in this situation.

In this paper, we generalize our previous results on generating common randomness at two terminals to a situation where any finite number of agents, interconnected by an arbitrary network of independent, point-to-point, discrete memoryless channels, wish to generate common randomness by interactive communication over the network. Our main result is an exact characterization of the common randomness capacity of such a network, i.e., the maximum number of bits of randomness that all the agents can agree on per step of communication. As a by-product, we also obtain a purely combinatorial result, viz., a characterization of (the incidence vectors of) the spanning arborescences rooted at a specified vertex in a digraph, and having exactly one edge exiting the root, as precisely the extreme points of a certain unbounded convex polyhedron, described by a system of linear inequalities.

We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of complexity classes initiated by Babai, Frankl, and Simon [Proc. 27th IEEE FOCS 1986, pp. 337--347] and continued in several papers (e.g., Halstenberg and Reischuk [Journ. of Comput. and Syst. Sci. 41(1990), pp. 402--429], Karchmer et al. [Journ. of Comput. and Syst. Sci. 49(1994), pp. 247--257] More precisely, it will be shown that the communication complexity classes MOD p P cc and MOD q P cc are incomparable with regard to inclusion, for all pairs of distinct prime numbers p and q. The same is true for PP cc and MODmP cc , for any number m 2. Moreover, nondeterminism and modularity are incomparable to a large extend. On the other hand, if m = p l 1 1 \Delta : : : \Delta p l r r ...

We prove that the modular communication complexity of the undirected graph connectivity problem UCONN equals \Theta(n), in contrast to the well-- known \Theta(n log n) bound in the deterministic case, and to the\Omega\Gamma n log log n) lower bound in the nondeterministic case, recently proved by Raz and Spieker. We obtain our result by combining Mobius function techniques due to Lovasz and Saxe with rank and projection reduction arguments.