It is known for many algorithmic problems that if a tree decomposition of width $t$ is given in the input, then the problem can be solved with exponential dependence on $t$. A line of research by Lokshtanov, Marx, and Saurabh [SODA 2011] produced lower bounds showing that in many cases known algorithms achieve the best possible exponential dependence on $t$, assuming the SETH. The main message of our paper is showing that the same lower bounds can be obtained in a more restricted setting: a graph consisting of a block of $t$ vertices connected to components of constant size already has the same hardness as a general tree decomposition of width $t$. Formally, a $(\sigma,\delta)$-hub is a set $Q$ of vertices such that every component of $Q$ has size at most $\sigma$ and is adjacent to at most $\delta$ vertices of $Q$. $\bullet$ For every $\epsilon> 0$, there are $\sigma,\delta> 0$ such that Independent Set/Vertex Cover cannot be solved in time $(2-\epsilon)^p\cdot n$, even if a $(\sigma,\delta)$-hub of size $p$ is given in the input, assuming the SETH. This matches the earlier tight lower bounds parameterized by the width of the tree decomposition. Similar tight bounds are obtained for Odd Cycle Transversal, Max Cut, $q$-Coloring, and edge/vertex deletions versions of $q$-Coloring. $\bullet$ For every $\epsilon>0$, there are $\sigma,\delta> 0$ such that Triangle-Partition cannot be solved in time $(2-\epsilon)^p\cdot n$, even if a $(\sigma,\delta)$-hub of size $p$ is given in the input, assuming the Set Cover Conjecture (SCC). In fact, we prove that this statement is equivalent to the SCC, thus it is unlikely that this could be proved assuming the SETH. $\bullet$ For Dominating Set, we can prove a non-tight lower bound ruling out $(2-\epsilon)^p\cdot n^{O(1)}$ algorithms, assuming either the SETH or the SCC, but this does not match the $3^p\cdot n^{O(1)}$ upper bound.

翻译：已知许多算法问题在输入中给出宽度为$t$的树分解时，可以以指数依赖$t$的方式求解。Lokshtanov、Marx和Saurabh [SODA 2011] 的研究路线给出了下界，表明在假设SETH成立的情况下，许多问题的已知算法已实现最佳的指数依赖$t$。本文的主要信息是表明，在更受限的设置中也能得到相同的下界：一个由$t$个顶点组成的块连接若干常数大小的分量构成的图，已具有与宽度为$t$的一般树分解相同的难解性。形式上，一个$(\sigma,\delta)$-枢纽是一个顶点集$Q$，使得$Q$的每个分量大小至多为$\sigma$，且与$Q$中至多$\delta$个顶点相邻。 $\bullet$ 对于任意$\epsilon>0$，存在$\sigma,\delta>0$，使得即使在输入中给出大小为$p$的$(\sigma,\delta)$-枢纽，独立集/顶点覆盖问题也无法在时间$(2-\epsilon)^p\cdot n$内求解（假设SETH成立）。这匹配了先前以树分解宽度为参数的紧下界。类似紧界在奇环横贯、最大割、$q$-染色以及$q$-染色的边/顶点删除版本中成立。 $\bullet$ 对于任意$\epsilon>0$，存在$\sigma,\delta>0$，使得即使在输入中给出大小为$p$的$(\sigma,\delta)$-枢纽，三角形划分问题也无法在时间$(2-\epsilon)^p\cdot n$内求解（假设集合覆盖猜想SCC成立）。事实上，我们证明该命题等价于SCC，因此不太可能基于SETH证明该结论。 $\bullet$ 对于控制集问题，我们可证明一个非紧下界，在假设SETH或SCC成立时排除了$(2-\epsilon)^p\cdot n^{O(1)}$算法，但这未能匹配$3^p\cdot n^{O(1)}$的上界。