We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with an unknown cost vector $c$ lying in a prior set $\mathcal{C}$. Recent work by Bennouna et al. provides an exact geometric characterization of sufficient decision datasets (SDDs) via an intrinsic decision-relevant dimension $d^\star$. However, their algorithm for constructing minimum-size SDDs requires solving mixed-integer programs. In this paper, we establish hardness results showing that computing $d^\star$ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving a recent open problem posed by Bennouna et al. To address this worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. Under nondegeneracy, we provide a polynomial-time cutting-plane algorithm for constructing pointwise-sufficient decision datasets. In a data-driven regime with i.i.d.\ costs, we further propose a cumulative algorithm that aggregates decision-relevant directions across samples, yielding a stable compression scheme of size at most $d^\star$. This leads to a distribution-free PAC guarantee: with high probability over the training sample, the pointwise sufficiency failure probability on a fresh draw is at most $\tilde{O}(d^\star/n)$, and this rate is tight up to logarithmic factors. Finally, we apply decision-sufficient representations to contextual linear optimization, obtaining compressed predictors with generalization bounds scaling as $\tilde{O}(\sqrt{d^\star/n})$ rather than $\tilde{O}(\sqrt{d/n})$, where $d$ is the ambient cost dimension.

翻译：我们研究如何构建压缩数据集，以充分恢复未知成本向量$c$位于先验集合$\mathcal{C}$中的线性规划问题的最优决策。Bennouna等人近期工作通过内在决策相关维度$d^\star$给出了充分决策数据集（SDDs）的精确几何特征。然而，其构建最小规模SDDs的算法需要求解混合整数规划。本文证明计算$d^\star$是NP难的，且判定数据集全局充分性是coNP难的，从而解决了Bennouna等人提出的近期开放问题。为应对这种最坏情况下的难解性，我们引入点态充分性——一种仅要求对单个成本向量具有充分性的松弛概念。在非退化条件下，我们提出多项式时间的割平面算法来构建点态充分决策数据集。在独立同分布成本的数驱动机制下，进一步提出累积算法，该算法跨样本聚合决策相关方向，生成规模至多为$d^\star$的稳定压缩方案。这引出了无关分布的概率近似正确保证：以高概率而言，训练样本上新鲜抽取点的点态充分性失效概率至多为$\tilde{O}(d^\star/n)$，且该速率在忽略对数因子下紧致。最后，我们将决策充分表示应用于上下文线性优化，获得泛化界为$\tilde{O}(\sqrt{d^\star/n})$而非$\tilde{O}(\sqrt{d/n})$的压缩预测器，其中$d$为环境成本维度。