Treewidth is as an important parameter that 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 fixed-parameter tractable parameterized by the treewidth of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], respectively. The algorithms generated by these (meta-)results have running times whose dependence on treewidth 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, exhibit the extraordinary computational hardness of such problems, and are rare: there are very few (for treewidth tw and vertex cover vc parameterizations) and they are for $\Sigma_2^p$-, $\Sigma_3^p$- or #NP-complete problems. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. Specifically, for the well-studied NP-complete metric graph 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, this 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, thereby adding to the short list of problems admitting such lower bounds. The latter results also yield lower bounds on the vertex-kernel sizes. We complement all our lower bounds with matching upper bounds.

翻译：树宽是一个能带来诸多问题可解性的重要参数。例如，可用一元二阶逻辑（MSO）表达及量化SAT（或更一般地，量化CSP）表达的图问题，分别以输入（原始）图的树宽加上MSO公式长度[Courcelle, Information & Computation 1990]和量词秩[Chen, ECAI 2004]为参数时是固定参数可解的。这些（元）结果生成的算法运行时间中，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问题，该指数塔的高度等于量词交替次数。运行时间至少需要双重指数因子的下界揭示了此类问题的非凡计算难度，且极为罕见：目前仅有极少数问题（针对树宽tw和顶点覆盖vc参数化）属于Σ₂ᵖ、Σ₃ᵖ或#NP完全问题。我们首次证明，无需上升至多项式层级更高阶即可获得此类下界。具体而言，对于已被充分研究的NP完全度量图问题——度量维数、强度量维数和测地集，我们证明即便在有限直径图上，它们也不存在2^{2^{o(tw)}}·n^{O(1)}时间算法（除非ETH失败）。对于强度量维数，该下界在vc参数化下依然成立。由于另外两个问题存在2^{O({vc}^2)}·n^{O(1)}时间算法，此类下界对它们不可行。我们证明，除非ETH失败，它们不存在2^{o({vc}^2)}·n^{O(1)}时间算法——从而拓展了拥有此类下界的极少数问题清单。后者结果还给出了顶点核大小的下界。我们为所有下界匹配了上界。