Structural Parameterization of Locating-Dominating Set and Test Cover

Abstract

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 $\textsf{NP} \subseteq \textsf{coNP}/poly$.

Figures (5)

• Figure 1: Hasse diagram of graph parameters and associated results for Locating-Dominating Set. An edge from a lower parameter to a higher parameter indicates that the lower one is upper bounded by a function of the higher one. If the line is dashed, then the bound is exponential; otherwise, it is polynomial. Colors correspond to the known FPT complexity status with respect to the highlighted parameter: the upper half of the box represents the upper bound, and the lower half of the box represents the lower bound. By "single-exponential", "slightly super-exponential" and "double-exponential", we mean functions of the form $2^{\mathcal{O}(p)}$, $2^{\mathcal{O}(p\log p)}$ or $2^{\mathcal{O}(p^2)}$, and $2^{2^{\mathcal{O}(p)}}$, respectively. The parameters for which the running time is not known to be tight are thus striped. The red circle in the upper-right corner means that Locating-Dominating Set does not admit a polynomial kernel when parameterized by the marked parameter unless $\NP\subseteq \coNP/poly$; the yellow one means that a (tight) quadratic kernel exists, and the green one, that a linear kernel exists. The bold borders highlight parameters that are covered in this paper.
• Figure 2: An instance $(G,k)$ of Locating-Dominating Set. Set $U$ is a minimum-size vertex cover of $G$. The dotted edge denotes that a vertex is adjacent with all the vertices in the set. For the sake of brevity, we do not show all the edges. The vertices in green ellipses are part of solution because of $(i)$ being a part of false-twins, $(ii)$ guessed intersection with $U$, and $(iii)$ the requirement of solution to be a dominating set. Note that the vertices $T_L$ are not dominated by the partial solution $Y_L$. Parts $P_1, P_2, P'_1, P'_2,$ and $P"_1$ are parts of partition of $R \cup B$ induced by $Y_L$.
• Figure 3: Instance of the Annotated Red-Blue Partition Refinement problem. Vertices in green ellipse denotes set $C_0$ whereas dotted ellipse denote vertices in $T_L$, and the partition $\mathcal{Q} = \{P_1, P_2, P'_1, P'_2, P"_1\}$. The vertices in the rectangles denote vertices in $T^{\circ}_L$.
• Figure 4: An illustrative example of the graph constructed by the reduction used in Section \ref{['sec:incompressibility']}. Adjacency to $y_{1,1}$, $y_{2, 1}$, $y_{3, 1}$ correspond to $1$ in the bit representation from left to right (most significant to least significant) bits. In the above example, the bit representation of $r_1$ is $\langle 1, 0, 0\rangle$, $r_2$ is $\langle 0, 1, 0\rangle$, $r_3$ is $\langle 1, 1, 0\rangle$, etc. For brevity, we only show the edges incident on $r_3$. The dotted line represents a non-edge. In this example, $r'_3$ is only adjacent to $b'_2$ in $G'$. It is easy to see that $r_3$ can only resolve pair $(b^{\circ}_2, b^{\star}_2)$.
• Figure 5: An illustrative example of the graph constructed by the reduction for Test Cover. Red (squared) nodes denote the tests whereas blue (filled circle) nodes the items. The adjacencies across $B$ and $\texttt{bit-rep}(B)$ are the same as the ones from the reduction for Locating-Dominating Set. For brevity, we only show the edges incident on $r_3$. The dotted line represents a non-edge. In this example, test corresponding to $r_3$ only contains item corresponding to $b_2$. It is easy to see that $r_3$ can only distinguish between the pair $(b^{\circ}_2, b^{\star}_2)$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 6
• Definition 7: Partition Induced by $C$ and Refinement
• Lemma 8
• Claim 9
• Lemma 10
• Corollary 11
• Proposition 12: KK20