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. This is impossible for the other two as they admit $2^{O({vc}^2)} \cdot n^{O(1)}$-time algorithms. We show that, unless the ETH fails, they do not admit $2^{o({vc}^2)}\cdot n^{O(1)}$-time algorithms, which are also rare results. The lower bounds for the vc parameterizations also yield rare lower bounds on the vertex-kernel sizes of these problems. We complement our lower bounds with matching upper bounds.

翻译：树宽（tw）是重要的参数，当其有界时能使许多问题具有可解性。例如，可用一元二阶（MSO）逻辑表达以及QUANTIFIED SAT（或更一般的QUANTIFIED CSP）等图问题，分别在输入（素）图的tw加MSO公式长度[Courcelle, Information & Computation 1990]和量词秩[Chen, ECAI 2004]参数化下属于固定参数可解（FPT）。这些（元）结果中的算法运行时间对tw的依赖呈指数塔形式。Fichte等人[LICS 2020]的条件性下界表明，对于QUANTIFIED SAT，该指数塔的高度等于量词交替次数。证明运行时间至少需要双指数因子的下界十分罕见：目前仅有极少数结果（针对tw和顶点覆盖参数化），且局限于#NP、$\Sigma_2^p$、$\Pi_2^p$或更高层多项式层次结构的完全问题。我们首次证明，无需上升到多项式层次结构更高层即可获得此类下界。我们基于Sperner族设计了一种新颖且通用的技术来建立此类下界，并将其应用于三个问题：度量维数、强度量维数和测地集。我们证明，除非ETH失效，否则这三个问题在直径有界图上均不存在$2^{2^{o(tw)}} \cdot n^{O(1)}$时间的算法。对于强度量维数，该下界在顶点覆盖参数化下同样成立。而其他两个问题因存在$2^{O({vc}^2)} \cdot n^{O(1)}$时间的算法，故无法获得相同下界。我们进一步证明，除非ETH失效，否则它们不存在$2^{o({vc}^2)}\cdot n^{O(1)}$时间的算法——此类结果同样罕见。针对顶点覆盖参数化的下界还导出了这些问题顶点核规模的罕见下界。我们通过匹配的上界补充了这些下界。