Resolve NP-completeness of computing communication complexity in Yao’s original alternating model

Determine whether deciding CC_alt(f) ≤ k is NP-hard, where CC_alt(f) denotes the deterministic communication complexity under Yao’s original alternating-round model (the smallest number of rounds/bits in an alternating protocol computing a Boolean function f given by its communication matrix). Together with the standard inclusion in NP, this would establish NP-completeness for computing CC_alt(f) and resolve Yao’s original question in the alternating model.

Background

The paper proves NP-hardness for computing CC(f) under the modern protocol-tree definition (minimum depth of a deterministic protocol tree). Yao’s original definition measures the smallest number of rounds (equivalently, bits) in an alternating protocol, and the two measures coincide up to a factor of 2.

The authors’ reduction is not robust to a constant-factor distortion, so their NP-hardness result for protocol-tree depth does not automatically imply NP-hardness for the alternating model. Since the decision problem “CC(f) ≤ k” is in NP, establishing NP-hardness for the alternating model would yield NP-completeness and fully answer Yao’s original question.