Tighten the upper bound on the complexity of planar p-levels

Determine whether the number of breakpoints in a planar p-level, and consequently in the cardinality-constrained incentive-allocation signature function, admits an upper bound substantially smaller than the currently known O(mp^{1/3}) bound.

Background

For a cardinality-constrained agent with m incentives and an allocation limit of p incentives, the signature function is determined by the p-level of an arrangement of m lines. The paper applies the best cited upper bound, O(mp{1/3}), on the number of p-level breakpoints to bound the complexity of the signature function and the resulting trade-off curve.

The paper notes that this upper bound may be far from tight: the currently known lower bound is only m 2{Ω(√log p)}. Improving the upper bound would directly improve the complexity guarantees for computing and maintaining the incentive-allocation trade-off curve.

References

The true upper bound for number of breakpoints in p-level might be much smaller. The currently known lower bound is only $m 2{\Omega(\sqrt{\log p})}$ .

— Large-Scale Trade-Off Curve Computation for Incentive Allocation with Cardinality and Matroid Constraints  (2609.20699 - Cong et al., 17 Sep 2026) in Section 3, subsection “Cardinality Constraint”

However, the running time is impractical for implementations and currently no nontrivial upper bound is known for the number of breakpoints on $f_i$.

— Large-Scale Trade-Off Curve Computation for Incentive Allocation with Cardinality and Matroid Constraints  (2609.20699 - Cong et al., 17 Sep 2026) in Section 4, “Submodular Objective”