Classify exact optimization for unrestricted market size with fixed support size at least three

Classify the computational complexity of computing and constructing optimal randomized dominant-strategy incentive-compatible, ex-post individually rational mechanisms for additive buyers with values drawn independently from a common finite distribution of fixed support size K≥3, when both the number of buyers and the number of items are unrestricted.

Background

The paper establishes polynomial-time exact optimization when any two of the buyer count, item count, and support size are fixed. It also gives an explicit polynomial-time optimum for two-point distributions, while proving #P-hardness for general finite-support distributions in two regimes: one buyer with an unrestricted number of items and two items with an unrestricted number of buyers.

The remaining regime in which the support size is fixed at K≥3 while both market dimensions may grow is not resolved by these results. The authors explicitly note that this case is unclassified; moreover, the existing exact hardness results do not rule out a fully polynomial-time approximation scheme.

References

Fixed K\ge3 with unrestricted n,m remains unclassified here, and exact hardness does not exclude an FPTAS.

— Optimal Multi-Item Auctions with I.I.D. Values  (2610.11051 - Liu et al., 8 Oct 2026) in Section 1, subsection “Our results and proof ideas” (following Table 1)