Tight (Double) Exponential Bounds for NP-Complete Problems: Treewidth and Vertex Cover Parameterizations
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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 Σ_2^p-, Σ_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)· 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)· n^O(1)-time algorithms. We show that, unless the ETH fails, they do not admit 2^o(vc^2)· 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.
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