🤖 AI Summary
This paper investigates the parameterized time lower bounds for three classical NP-complete graph problems—Metric Dimension, Strong Metric Dimension, and Geodetic Set—with respect to treewidth (tw) and vertex cover (vc). Under the Strong Exponential Time Hypothesis (SETH), we establish, for the first time, that none of these problems admits a $2^{2^{o(tw)}} cdot n^{O(1)}$-time algorithm, even when the input graph has bounded diameter; moreover, Strong Metric Dimension also lacks a $2^{2^{o(vc)}} cdot n^{O(1)}$-time algorithm. This yields the first double-exponential lower bounds for problems in NP, challenging the conventional belief that only problems beyond the polynomial hierarchy require double-exponential time. Technically, we introduce a generic construction based on Sperner families, integrating ETH-based reductions with precise parameterized complexity analysis. We complement our lower bounds with matching upper bounds, thereby fully characterizing the double-exponential hardness of these problems.
📝 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, $Sigma_2^p$, $Pi_2^p$, 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.
