Consider a finite ground set $E$, a set of feasible solutions $X \subseteq \mathbb{R}^{E}$, and a class of objective functions $\mathcal{C}$ defined on $X$. We are interested in subsets $S$ of $E$ that control $X$ in the sense that we can induce any given solution $x \in X$ as an optimum for any given objective function $c \in \mathcal{C}$ by adding linear terms to $c$ on the coordinates corresponding to $S$. This problem has many applications, e.g., when $X$ corresponds to the set of all traffic flows, the ability to control implies that one is able to induce all target flows by imposing tolls on the edges in $S$. Our first result shows the equivalence between controllability and identifiability. If $X$ is convex, or if $X$ consists of binary vectors, then $S$ controls $X$ if and only if the restriction of $x$ to $S$ uniquely determines $x$ among all solutions in $X$. In the convex case, we further prove that the family of controlling sets forms a matroid. This structural insight yields an efficient algorithm for computing minimum-weight controlling sets from a description of the affine hull of $X$. While the equivalence extends to matroid base families, the picture changes sharply for other discrete domains. We show that when $X$ is equal to the set of $s$-$t$-paths in a directed graph, deciding whether an identifying set of a given cardinality exists is $Σ\mathsf{_2^P}$-complete. The problem remains $\mathsf{NP}$-hard even on acyclic graphs. For acyclic instances, however, we obtain an approximation guarantee by proving a tight bound on the gap between the smallest identifying sets for $X$ and its convex hull, where the latter corresponds to the $s$-$t$-flow polyhedron.

翻译：考虑一个有限基集$E$、一组可行解$X \subseteq \mathbb{R}^{E}$，以及定义在$X$上的一类目标函数$\mathcal{C}$。我们关注$E$的子集$S$对$X$的控制能力：即通过对$c$在$S$对应坐标上添加线性项，能够使任意给定目标函数$c \in \mathcal{C}$诱导出任意给定解$x \in X$作为最优解。该问题具有广泛的应用背景，例如当$X$对应所有交通流集合时，控制能力意味着通过在边集$S$上施加通行费即可诱导出所有目标流量。我们的首要结论揭示了可控性与可识别性之间的等价关系：若$X$为凸集，或$X$由二元向量构成，则$S$控制$X$当且仅当$x$在$S$上的限制能唯一确定$X$中的所有解。在凸情形下，我们进一步证明控制集族构成拟阵，这一结构特性使得从$X$的仿射包描述出发，能高效计算最小权控制集。虽然该等价关系可推广至拟阵基族，但在其他离散域中情况发生显著变化。我们证明当$X$等于有向图中的$s$-$t$路径集合时，判定给定基数的识别集是否存在属于$Σ\mathsf{_2^P}$完全问题，即使在无环图上该问题仍保持$\mathsf{NP}$困难性。然而对于无环实例，通过严格界定$X$与其凸包（对应$s$-$t$流多面体）的最小识别集之间的间隙，我们获得了近似保证。