SIAM Journal on Computing, Volume 49, Issue 6, Page 1291-1331, January 2020.
We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d\ge 2$ for many well-known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms are representation-agnostic, i.e., they work on the graph itself and do not require the geometric representation. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into $d$-dimensional grids, and it allows us to derive matching $2^{\Omega(n^{1-1/d})}$ lower bounds under the exponential time hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.

中文翻译：

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指数时间假说的框架-几何相交图中的紧算法和下界

SIAM计算杂志，第49卷，第6期，第1291-1331页，2020年1月。
我们给出了一个算法和下界框架，该框架有助于构建次指数算法并匹配条件复杂性范围。它可以应用于大小相似的胖对象的相交图，对于许多固定尺寸的对象，对于任何固定维数$ d \ ge 2 $，其运行时间为$ 2 ^ {O（n ^ {1-1 / d}）} $。已知的图形问题，包括独立集，常数$ r $的支配集和斯坦纳树。对于大多数问题，与以前的工作相比，我们可以缩短运行时间；在某些情况下，我们在几何相交图中给出了第一个已知的次指数算法。另外，大多数获得的算法都是与表示无关的，即，它们在图形本身上起作用，并且不需要几何表示。我们的算法框架基于加权分隔符定理和各种树宽技术。下界框架基于将图构造性地嵌入到$ d $维网格中，它允许我们导出指数下匹配的$ 2 ^ {\ Omega（n ^ {1-1 / d}）} $下界时间假设，即使在$ d $维诱导网格图的限制更为严格的类中也是如此。