Lower bound for Greedy in triangle-free-crossing matching

Prove or disprove that Greedy for online triangle-free-crossing matching has asymptotic competitive ratio at least 2/3.

Background

The paper proves that Greedy has asymptotic competitive ratio at most 2/3 in the triangle-free-crossing model and gives a different two-layer-Greedy algorithm with exact ratio 4/5. A matching lower bound for Greedy is identified as the principal unresolved issue. Through the blocking-number correspondence, the question is equivalent to determining whether every triangle-free-crossing family of ℓ segments blocks at most ℓ points.

References

For triangle-free-crossing matching Greedy has a competitive ratio of at most $2/3$, and we show this by presenting an elegant construction. Surprisingly, proving a matching lower bound seems difficult and is left as the main open problem of this work.

— Online geometric matching and segments block points  (2610.23130 - Li et al., 16 Oct 2026) in Section 1, Our contribution; formalized in Section 6, Open problems