In this paper, we present a discrepancy-based approach to cost-preserving rounding under hard constraints. Starting from a feasible fractional solution, our rounding procedure preserves the designated hard constraints, never increases the linear objective, and bounds the error in the remaining constraints by the discrepancy of a suitable column-difference matrix. Our second discrepancy-theoretic result bounds the discrepancy of a matrix in terms of a novel column parameter, called mountain distance. This notion captures ordered structural features of a matrix which are invisible to the classical Beck--Fiala and Komlós column-norm parameters. We show a variety of applications of these two results. In particular, we obtain new results for min-cost unsplittable flows, bounded-turn routing in grids and tori, rainbow prefix hypergraph decomposition, and $2$-multicast scheduling.

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