Unified conjecture combining sparsity, multi-winner columns, and nonuniform weights
Determine whether, for fractional matrices x ∈ [0,1]^{m×n} with integer column sums n_j := ∑_{i=1}^m x_{ij} ∈ ℕ and nonuniform weights d_{ij} > 0, there exists an integral assignment y ∈ {0,1}^{m×n} satisfying support constraints (y_{ij} = 0 whenever x_{ij} = 0) and 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_{ij} (x_{ij} − y_{ij})| ≤ max_{i ∈ [m], j ∈ [n]} d_{ij}.
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It is plausible that Conjectures~\ref{conj:weighted_carpool},~\ref{conj:non_uniform} and~\ref{conj:int_weights} may all be combined: Let m, n be positive integers and d \in R{m \times 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], y_{ij} = 0 whenever x_{ij} = 0, and
\Big|\sum_{j \in [t]} d_{ij} (x_{ij}-y_{ij}) \Big| \le \max_{i \in [m], j \in [n]} d_{ij}\quad \forall i\in[m], t\in [n].