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.

翻译：树宽 (tw) 是一个重要参数，当它被限定时，许多问题可得到可解性。例如，可用一元二阶 (MSO) 逻辑表达的图问题、量化 SAT 或更一般的量化 CSP，在输入（原始）图的树宽加上 MSO 公式长度 [Courcelle, Information & Computation 1990] 和量词秩 [Chen, ECAI 2004] 分别作为参数时，属于固定参数可解 (FPT) 问题。这些（元）结果中的算法，其运行时间对树宽的依赖呈指数塔形式。Fichte 等人 [LICS 2020] 的条件性下界表明，对于量化 SAT，该指数塔的高度等于量词交替的次数。表明运行时间中至少需要双指数因子的下界非常罕见：对于树宽和顶点覆盖 (vc) 参数化，仅有极少数结果，且它们针对的是 #NP、Σ₂ᵖ、Π₂ᵖ 或多项式层级更高层的完全问题。我们首次证明，无需上升到多项式层级的更高层即可得到此类下界。我们设计了一种基于 Sperner 族的全新且简单的通用技术来获得此类下界，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，否则即使在有界直径图上，这些问题也不存在 2^{2^{o(tw)}} · n^{O(1)} 时间的算法。对于强度量维数，该下界甚至对 vc 也成立。我们通过匹配的上界对下界进行了补充。