Establish NP-hardness of approximation for deterministic communication complexity CC(f)

Determine whether approximating CC(f), the deterministic communication complexity under the protocol-tree model given a function’s communication matrix, is NP-hard within any constant multiplicative factor greater than 1. Proving such an NP-hardness-of-approximation would make the hardness result robust to protocol rebalancing.

Background

The paper proves NP-hardness of computing CC(f) exactly but provides no hardness-of-approximation guarantee. Prior work shows cryptographic hardness for approximating CC(f) within about 1.1, yet no NP-hardness-of-approximation is known.

The authors emphasize that an NP-hardness result for approximation (ideally to a constant factor, e.g., 4) would yield robustness to the constant-factor differences between models and to rebalancing of protocol trees, but achieving this appears to require fundamentally new reductions.