Establish tightness of the general approximate-call bound

Establish whether the general path-cover approximation bound \(\frac{2\alpha+\beta-\alpha\beta}{4-\beta}\rho_w(G)\) for the weighted edge-deletion rule is tight for arbitrary matching approximation ratios \(\alpha\) and \(\beta\).

Background

The weighted edge-deletion algorithm computes an approximate matching in the original graph, an approximate matching after deleting the first matching, and then breaks cycles. The paper proves the bound 2α+β−αβ4−βρw(G)\frac{2\alpha+\beta-\alpha\beta}{4-\beta}\rho_w(G), and establishes tightness only for exact calls, where the ratio is 2/3. Whether the stated bound is tight for general approximation parameters remains unresolved.

References

This exact-call constant is tight (\Cref{prop:deletion-tight}); tightness of the general $(\alpha,\beta)$ bound is not established.

— Sublinear-Time and Semi-Streaming Algorithms for Path Cover and TSP  (2610.18701 - Alipour et al., 13 Oct 2026) in Theorem 2.3 discussion in Section 2, immediately after Theorem 2.3

The deletion rule's completed-tour infimum remains open.

— 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 “Semi-streaming construction”