Heterogeneous-cost structural bound for the greedy gap

Establish a heterogeneous-cost analogue of the equal-cost structural bound on the penalized greedy gap for the EIG-Cost policy, thereby controlling the greedy suboptimality term without exhaustive optimization.

Background

The paper proves a structural upper bound on the penalized greedy gap when all acquisition costs are equal. In that setting, the cost penalty is cardinality-dependent, so greedy acquisition retains the ordering of ordinary information-gain greedy selection and the submodularity-ratio analysis applies.

For heterogeneous costs, this cardinality reduction fails because different acquisition costs alter the ordering. The authors report that the greedy gap is nevertheless empirically near zero on tractable heterogeneous-cost instances, but explicitly leave open whether a corresponding theoretical bound exists.