# Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results

For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets σ,ρ of non-negative integers, a (σ,ρ)-set of a graph G is a set S of vertices such that |N(u)∩ S|∈σ for every u∈ S, and |N(v)∩ S|∈ρ for every v∉S. The problem of finding a (σ,ρ)-set (of a certain size) unifies common problems like Independent Set, Dominating Set, Independent Dominating Set, and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets (σ,ρ), there is an algorithm that counts (σ,ρ)-sets in time (c_σ,ρ)^tw· n^O(1) (if a tree decomposition of width tw is given in the input). Here, c_σ,ρ is a constant with an intricate dependency on σ and ρ. Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair (σ, ρ) of finite or cofinite sets where the problem is non-trivial, and any ε>0, a (c_σ,ρ-ε)^tw· n^O(1)-algorithm counting the number of (σ,ρ)-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

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