Metric Dimension and Geodetic Set Parameterized by Vertex Cover

TL;DR

This work analyzes Metric Dimension and Geodetic Set under the vertex cover parameter, establishing FPT algorithms with running time $2^{\mathcal{O}(\mathrm{vc}^2)}\cdot n^{\mathcal{O}(1)}$ and kernels of size $2^{\mathcal{O}(\mathrm{vc})}$, while proving ETH-based lower bounds ruling out $2^{o(\mathrm{vc}^2)}$-time algorithms and $2^{o(\mathrm{vc})}$-vertex kernels, even on bounded-diameter graphs. It introduces a versatile gadget toolkit—Set Identifying Gadget, Gadget to Add Critical Pairs, and Vertex Selector Gadgets—to build reductions encoding logical structure into metric-based constraints. The core technical contribution is an ETH-based reduction from 3-Partitioned-3-SAT that yields a graph $G$ with $|V(G)|=2^{O(\sqrt{N})}$ and $\mathrm{vc}(G) + k = O(\sqrt{N})$, thereby establishing tight exponential lower bounds for these problems parameterized by vertex cover. The methods extend to Geodetic Set and demonstrate a general approach for proving exponential lower bounds and kernel-size limits for metric/shortest-path problems under structural parameters, with potential applicability to additional problems in this domain.

Abstract

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

Figures (3)

• Figure 1: Set Identifying Gadget (left). The blue box represents $\textsf{bit-rep}(X)$. The yellow lines represent that all possible edges exist between $\textsf{bit-rep}(X)\setminus \textsf{bits}(X)$ and $\textsf{nullifier}(X)$, $\textsf{nullifier}(X)$ and $N(X)$, and $y_{\star}$ and $X$. Note that $G'$ is not necessarily restricted to the graph induced by the vertices in $X\cup N(X)$. Vertex Selector Gadget (right). For $X \in \{B, A\}$, the blue box represents $\textsf{bit-rep}(X)$, the blue link represents the connection with respect to the binary representation, and the yellow line represents that $\textsf{nullifier}(X)$ is adjacent to each vertex in $\textsf{bit-rep}(X)\setminus \textsf{bits}(X)$. Dotted lines highlight absent edges.
• Figure 2: Overview of the reduction. Sets in ellipses are independent sets and sets in rectangles are cliques. For $X \in \{B^{\alpha}, A^{\alpha}, P^{\alpha}, C\}$, the blue rectangle attached to it via the blue edge represents $\textsf{bit-rep}(X)$. We omit $\textsf{bits}(X)$ for legibility. The yellow line represents that $\textsf{nullifier}(X)$ is connected to every vertex in the set. Note the exception of $\textsf{nullifier}(P^{\alpha})$ which is not adjacent to any vertex in $A^{\alpha}$. Purple lines between two sets denote adjacencies with respect to indexing, i.e., $b^{\alpha, \circ}_i$ is adjacent only with all the vertices in $A^{\alpha}_{i}$, and all the vertices in $A^{\alpha}_i$ are adjacent with $v^{\alpha}_i$ in validation portal $V^{\alpha}$. Gray lines also indicate adjacencies with respect to indexing, but in a complementary way. If $C_q$ contains a variable in $B^{\alpha}_i$, then $c^{\circ}_q$ and $c^{\star}_q$ are adjacent with all vertices $v^{\alpha}_j \in V^{\alpha}$ such that $j \neq i$. Green and red lines between the $A^{\alpha}$ and $T^{\alpha}$ and $F^{\alpha}$ roughly transfer, for each $a_{i,\ell}^{\alpha}\in A^{\alpha}$, the underlying assignment structure. If the $j^{th}$ variable by $a_{i,\ell}^{\alpha}$ is assigned to True, then we add the green edge $(a_{i,\ell}^{\alpha}, t_{j}^{\alpha})$, and otherwise the red edge $(a_{i,\ell}^{\alpha}, f_{j}^{\alpha})$. Similarly, we add edges for each $c_i^{\circ}\in C$ depending on the assignment satisfying the variable from the part $X^{\delta}$ of a clause $c_i$, and in which block $B_j^{\delta}$ it lies, putting either an edge $(c_i^{\circ},t_j^{\delta})$ or $(c_i^{\circ},f_j^{\delta})$ accordingly ($\delta \in \{\alpha, \beta, \gamma\}$).
• Figure 3: A toy example to illustrate the core ideas in the reduction. Note that $\textsf{bit-rep}$ and $\textsf{nullifier}$ are omitted for the sets.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Proposition 5: DBLP:journals/corr/abs-2302-09604
• Theorem 6
• Corollary 7
• Corollary 8
• Lemma 9