Weighted committee assignment
Establish whether, for fractional matrices x ∈ [0,1]^{m×n} with integer column sums n_j := ∑_{i=1}^m x_{ij} ∈ ℕ and weights d_1, ..., d_n > 0, there exists an integral assignment y ∈ {0,1}^{m×n} respecting the column sums (∑_{i=1}^m y_{ij} = n_j for all j) such that, for all rows i ∈ [m] and prefixes t ∈ [n], the weighted prefix discrepancy satisfies |∑_{j=1}^t d_j (x_{ij} − y_{ij})| ≤ max_{j ∈ [n]} d_j.
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We also formulate a weighted committee assignment problem: Let m, n be positive integers and d \in R{n}_{>0}. For every x \in [0,1]{m \times n} so that n_j := \sum_{i \in [m]} x_{ij} \in \mathbb{N} for all j \in [n], there is y \in {0,1}{m \times n} so that \sum_{i \in [m]} y_{ij} = n_j for all j \in [n] and
\Big|\sum_{j \in [t]} d_{j} (x_{ij}-y_{ij}) \Big| \le \max_{j \in [n]} d_{j}\quad \forall i\in[m], t\in [n].