Optimality of the Pass and Space Bounds

Determine whether the pass count and multi-pass semi-streaming space bound for estimating the Held–Karp value are optimal, and whether the required pass count must grow with the number of vertices.

Background

The paper establishes a multi-pass semi-streaming algorithm for estimating the Held–Karp bound and proves that, on weighted graph streams, every one-pass approximation with a fixed factor below two requires quadratic space. However, the lower bound does not match the algorithm’s multi-pass resource guarantees. In particular, the authors leave unresolved whether the stated pass complexity or space usage can be improved, and whether any nonconstant dependence of the pass complexity on the number of vertices is fundamentally necessary.