Determine the optimal retention coefficient and running-time exponent

Determine the best possible retention coefficient and running-time exponent for sublinear-time path-cover estimation, beyond the limitations supplied by the quadratic-query lower bound and the computational restriction discussed in the paper.

Background

The paper develops a sublinear-time estimator for maximum path-cover size based on a randomly ordered implicit bipartite representation. Its analysis achieves the coefficient 2(sec 1 − tan 1) and a subquadratic running time, while cited lower bounds exclude near-optimal multiplicative estimation in truly subquadratic time. The authors explicitly state that these bounds do not identify the optimal tradeoff between the retained fraction of the optimum and the running-time exponent.

References

Neither bound determines the best retention coefficient or running-time exponent.

— Sublinear-Time and Semi-Streaming Algorithms for Path Cover and TSP  (2610.18701 - Alipour et al., 13 Oct 2026) in Section 4, subsection “Open questions,” paragraph “Sublinear-time estimation”