We consider the following communication problem: Alice and Bob each have some
valuation functions v1β(β ) and v2β(β ) over subsets of m items,
and their goal is to partition the items into S,SΛ in a way that
maximizes the welfare, v1β(S)+v2β(SΛ). We study both the allocation
problem, which asks for a welfare-maximizing partition and the decision
problem, which asks whether or not there exists a partition guaranteeing
certain welfare, for binary XOS valuations. For interactive protocols with
poly(m) communication, a tight 3/4-approximation is known for both
[Fei06,DS06].
For interactive protocols, the allocation problem is provably harder than the
decision problem: any solution to the allocation problem implies a solution to
the decision problem with one additional round and logm additional bits of
communication via a trivial reduction. Surprisingly, the allocation problem is
provably easier for simultaneous protocols. Specifically, we show:
1) There exists a simultaneous, randomized protocol with polynomial
communication that selects a partition whose expected welfare is at least 3/4
of the optimum. This matches the guarantee of the best interactive, randomized
protocol with polynomial communication.
2) For all Ξ΅>0, any simultaneous, randomized protocol that
decides whether the welfare of the optimal partition is β₯1 or β€3/4β1/108+Ξ΅ correctly with probability >1/2+1/poly(m) requires
exponential communication. This provides a separation between the attainable
approximation guarantees via interactive (3/4) versus simultaneous (β€3/4β1/108) protocols with polynomial communication.
In other words, this trivial reduction from decision to allocation problems
provably requires the extra round of communication