Weighted carpooling prefix discrepancy

Establish whether, for every fractional assignment x ∈ [0,1]^{m×n} and weights d_1, ..., d_n > 0, there exists an integral assignment y ∈ {0,1}^{m×n} respecting sparsity (y_{ij} = 0 whenever x_{ij} = 0) such that, for all rows i ∈ [m] and all prefixes t ∈ [n], the weighted prefix discrepancy satisfies |∑_{j=1}^t d_j (x_{ij} − y_{ij})| ≤ max_{j ∈ [n]} d_j.

Background

The carpooling problem constrains assignments to the support of x (an item j can only be assigned to i if x_{ij} ≠ 0). The weighted version remains open and can be reduced to the single-source unsplittable flow discrepancy conjecture via a similar construction connecting terminals to permissible paths.

Known bounds include a linear-algebraic 2m·max_j d_j discrepancy and an O(√log n) bound via Banaszczyk’s convex-geometric argument, showing the LP relaxation for maximum flow-time has integrality gap O(√log n). The paper’s Proposition 2 indicates that a (1−δ)·max_j d_j bound is impossible even for m=3 under the sparsity constraint.

References

A variant of the chairman assignment problem is the carpooling problem, where j\in[n] can be assigned to i\in [m] if and only if x_{ij}\neq 0. The weighted carpooling problem, which remains open, can also be viewed as a special case of Conjecture \ref{conj:ssuf} using a similar construction that connects terminal t_j to paths i for which x_{ij}\neq 0: Let m, n be positive integers and d \in Rn_{>0}. For every fractional assignment x \in [0,1]{m \times n}, there is an assignment y \in {0,1}{m \times n} so that y_{ij} = 0 whenever x_{ij} = 0, and for every i\in [m] and t\in [n], \Big|\sum_{j \in [t]} d_j (x_{ij}-y_{ij}) \Big| \le \max_{j \in [n]} d_j.

— Weighted Chairman Assignment and Flow-Time Scheduling (2511.18546 - Liu et al., 23 Nov 2025) in Conjecture 2, Section 2 (Related work)