Problems in NP can Admit Double-Exponential Lower Bounds when Parameterized by Treewidth or Vertex Cover

TL;DR

The paper establishes ETH-based double-exponential lower bounds for Metric Dimension, Geodetic Set, and Strong Metric Dimension when parameterized by treewidth or vertex cover, even on graphs of bounded diameter. It introduces Sperner-family inspired gadgets, notably Set Identifying and Set Representation gadgets, to encode SAT constraints while preserving small separators and favorable width parameters. The main results show that these problems cannot be solved in time $2^{2^{o(tw)}}\cdot n^{O(1)}$ (and analogous bounds for vc) and complement these lower bounds with matching upper bounds, highlighting the necessity of tower-type dependencies in certain parameter regimes. The techniques, including the bit-representation and separator-based encodings, offer a versatile framework potentially applicable to other problems in structural parameterized complexity. Overall, the work reframes the landscape of tractability for metric graph problems by demonstrating intrinsic double-exponential barriers under ETH for key structural parameters.

Abstract

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $Σ_2^p$, $Π_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

Figures (7)

• Figure 1: Graph representations of 3-Partitioned-3-SAT. (Left) incidence graph representation. (Right) representation with small separators using our technique. Note, for example, that $x_1^{\alpha}$ appears as a positive literal in the clause $C_1$. Thus, on the left, $t_2^{\alpha}$ is the only literal vertex in $A^{\alpha}$ incident to $c_1$, while on the right, $t_2^{\alpha}$ is the only literal vertex in $A^{\alpha}$ that does not share a common neighbor with $c_1$ in $V^{\alpha}$. The edges from $c_2$ to each vertex in $V^{\alpha}$ are omitted for clarity.
• Figure 2: Set Identifying Gadget. The blue box represents $\textsf{bit-rep}(X)$ and the yellow lines represent that $\textsf{nullifier}(X)$ is adjacent to each vertex in $(\textsf{bit-rep}(X)\setminus \textsf{bits}(X))\cup N(X)$, and $y_{\star}$ is adjacent to each vertex in $X$. Also, $G'$ is not necessarily restricted to the graph induced by the vertices in $X\cup N(X)$.
• Figure 3: Set Representation Gadget. Let $\phi(q) = i$, i.e., only $a_i$ in $A$ can resolve the critical pair $\langle c^{\circ}_q, c^{\star}_q \rangle$. Let the vertices in $V$ be indexed from top to bottom and let $\textsf{set-rep}(i) = \{2, 4, 5\}$. By construction, the only vertices in $V$ that $c^{\star}_q$ is not adjacent to are $v_2$, $v_4$, and $v_5$ (this is highlighted by red-dotted edges). Thus, $\mathop{\mathrm{dist}}\nolimits(a_i, c^{\circ}_q) = 2$ and $\mathop{\mathrm{dist}}\nolimits(a_i, c^{\star}_q) > 2$, and hence, $a_i$ resolves $\langle c^{\circ}_q, c^{\star}_q \rangle$. For any other vertex in $A$, say $a_j$, $\textsf{set-rep}(j) \setminus \textsf{set-rep}(i)$ is non-empty, and thus, $a_j$ cannot resolve $\langle c^{\circ}_q, c^{\star}_q \rangle$.
• Figure 4: Reduction for proof of Theorem \ref{['thm:lower-bound-diam-tw']}. For any $X\in \{X^{\alpha},A^{\alpha},V^{\alpha},C\}$, the yellow line between a vertex and a blue box containing $\textsf{bit-rep}(X)$ indicates that vertex is connected to every vertex in $\textsf{bit-rep}(X)\setminus \textsf{bits}(X)$. The remainder of the yellow lines represent that vertex is connected to every vertex in the set the edge goes to. Green edges denote adjacencies with respect to $\textsf{set-rep}$, e.g., $t^{\alpha}_{2i}$ is adjacent to $v_j\in V^{\alpha}$ if $j\in \textsf{set-rep}(2i)$. Purple lines also indicate adjacencies with respect to $\textsf{set-rep}$, but in a complementary way, i.e., if $x_i\in c_q$, then, for every $p'\in [2p]\setminus \textsf{set-rep}(2i)$, we have $(v_{p'}^{\alpha}, c_q^{\star})\in E(G)$, and if $\overline{x}_i\in c_q$, then, for all $p'\in [2p]\setminus \textsf{set-rep}(2i-1)$, we have $(v_{p'}^{\alpha}, c_q^{\star})\in E(G)$.
• Figure 5: Set Identifying Gadget (left). The blue box represents $\textsf{bit-rep}(X)$ and the yellow lines represent that every vertex in $\textsf{bit-rep}(X)\setminus \textsf{bits}(X)$ is adjacent to $\textsf{nullifier}(X)$, $\textsf{nullifier}(X)$ is adjacent to every vertex in $N(X)$, and $y_{\star}$ is adjacent to every vertex in $X$. 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 connected to every vertex in $\textsf{bit-rep}(X)\setminus \textsf{bits}(X)$. The dotted lines highlight the absence of edges.

Theorems & Definitions (5)

• Theorem : \ref{['thm:lower-bound-diam-tw']}
• Proposition 4: DBLP:journals/corr/abs-2302-09604
• Proposition 5
• Theorem 6
• Lemma 7