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 $Q$. We show that $\bullet$ 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. Our results reveal that, for many problems, the research on lower bounds on the dependence on tree width was never really about tree decompositions, but the real source of hardness comes from a much simpler structure.

翻译：已知对于许多算法问题，若输入中给定宽度为$t$的树分解，则问题可按指数依赖于$t$的方式求解。Lokshtanov、Marx和Saurabh [SODA 2011] 的研究路线给出了下界，表明在许多情况下，基于SETH假设，已知算法已经达到了最优的指数依赖$t$。本文的主要结论是：相同下界可在更受限设定中获得——由$t$个顶点构成的块连接若干常数大小的组件，其困难程度已等价于宽度为$t$的通用树分解。形式化地，$(\sigma,\delta)$-枢纽指顶点集$Q$，使得$Q$的每个连通分量大小不超过$\sigma$，且与$Q$中至多$\delta$个顶点相邻。我们证明： $\bullet$ 对任意$\epsilon>0$，存在$\sigma,\delta>0$，使得即便输入中给定大小为$p$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（基于SETH假设）。这匹配了先前基于树分解宽度的紧下界。类似紧界也适用于奇环横向、最大割、$q$着色及$q$着色的边/顶点删除版本。 $\bullet$ 对任意$\epsilon>0$，存在$\sigma,\delta>0$，使得即便输入中给定大小为$p$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（基于集合覆盖猜想SCC）。事实上，我们证明该命题等价于SCC，因此难以基于SETH证明。我们的结果表明，对许多问题而言，关于树宽依赖性的下界研究从未真正聚焦于树分解，其难解性的真正根源来自一种更简单的结构。