Direct-sum width lower bound for k-parallel approximate counting in ROBP

Prove or disprove that computing the k-parallel approximate counting problem \(\approxcountp_k[n, n/3]\) in the read-once branching program (ROBP) model requires width \(\Omega(n)^k\) for integers \(n, k\) with \(n \gg k \ge 1\). Specifically, given an input stream \((x_1, \ldots, x_n) \in (\{0,1\}^k)^n\), determine whether any ROBP that outputs, for each coordinate \(j \in [k]\), a real estimate within additive error \(\le n/3\) of the number of 1s in \((x_1)_j, \ldots, (x_n)_j\) must have width at least \(\Omega(n)^k\).

Background

The paper proves that computing the k-parallel approximate counting problem $\approxcountp_k[n, n/3]$ requires width $n^{\Omega(k)}$ in the ROBP model (Theorem 5.1). This establishes a strong direct-sum-type lower bound in terms of bits of space, but it falls short of the conjectured tight $\Omega(n)^k$ width bound for fixed k.

The authors explain that their current technique, which tracks growth of a potential function across layers while excluding points on low-dimensional rectangle boundaries, cannot show $\Omega(n)^k$ growth. They therefore formulate an explicit open problem asking for a proof or disproof of the stronger $\Omega(n)^k$ width bound, which would constitute a tight direct-sum theorem for deterministic, worst-case ROBPs tackling k parallel instances of {0,1}-approximate counting.