We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if there is a subset $S$ of $k$ vertices such that any two distinct vertices not in $S$ are dominated by distinct subsets of $S$. In the second problem, an input is a set of items $U$, a set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

翻译：本文研究了两个识别问题的结构参数化：定位支配集（LOCATING-DOMINATING SET）与测试覆盖（TEST COVER）。在第一个问题中，输入是一个包含 $n$ 个顶点的图 $G$ 和一个整数 $k$，要求判断是否存在一个大小为 $k$ 的顶点子集 $S$，使得任意两个不在 $S$ 中的不同顶点被 $S$ 的不同子集所支配。在第二个问题中，输入是一个物品集合 $U$、一组称为测试的 $U$ 的子集 $\mathcal{F}$ 以及一个整数 $k$，要求判断是否存在一个至多包含 $k$ 个测试的集合 $S$，使得任意两个物品属于 $S$ 中测试的不同子集。这两个问题分别是支配集（DOMINATING SET）与集合覆盖（SET COVER）的“识别”版本。Chakraborty 等人 [ISAAC 2024] 证明了当以输入图的树宽为参数时，这两个问题均存在条件性双指数下界以及匹配的算法。我们延续这一研究方向，考虑比树宽更大的参数，如顶点覆盖数与反馈边集数。我们设计了一种非平凡的动态规划方案，以在物品数量 $|U|$ 上“略超指数”的时间 $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ 内求解测试覆盖问题，并在时间 $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$ 内求解定位支配集问题，其中 $\textsf{vc}$ 为顶点覆盖数，$n$ 为图的阶数。这表明 Chakraborty 等人 [ISAAC 2024] 中关于树宽的下界结果无法推广到顶点覆盖数。我们还证明了，以反馈边集数为参数时，定位支配集存在线性核，从而回答了 [Cappelle 等人, LAGOS 2021] 中的一个开放性问题。最后，我们证明除非 $\textsf{NP} \subseteq \textsf{coNP}/poly$，否则定位支配集与测试覆盖均不太可能存在能返回具有次平方比特数输入的压缩算法。