Problems in NP can Admit Double-Exponential Lower Bounds when Parameterized by Treewidth or Vertex Cover
arxiv(2023)
摘要
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, Σ_2^p, Π_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)· 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.
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