تقرير
Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results
العنوان: | Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results |
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المؤلفون: | Focke, Jacob, Marx, Dániel, Inerney, Fionn Mc, Neuen, Daniel, Sankar, Govind S., Schepper, Philipp, Wellnitz, Philip |
سنة النشر: | 2023 |
المجموعة: | Computer Science |
مصطلحات موضوعية: | Computer Science - Computational Complexity, Computer Science - Data Structures and Algorithms, F.2.2, G.2.1 |
الوصف: | 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 $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies common problems like $\text{Independent Set}$, $\text{Dominating Set}$, $\text{Independent Dominating Set}$, and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets $(\sigma,\rho)$, there is an algorithm that counts $(\sigma,\rho)$-sets in time $(c_{\sigma,\rho})^{\text{tw}}\cdot n^{O(1)}$ (if a tree decomposition of width $\text{tw}$ is given in the input). Here, $c_{\sigma,\rho}$ is a constant with an intricate dependency on $\sigma$ and $\rho$. Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\text{tw}}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis ($\#$SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well. Comment: This is the second part following an accompanying paper arXiv:2211.04278. We split the original paper to keep paper length more manageable |
نوع الوثيقة: | Working Paper |
URL الوصول: | http://arxiv.org/abs/2306.03640 |
رقم الأكسشن: | edsarx.2306.03640 |
قاعدة البيانات: | arXiv |
الوصف غير متاح. |