宽度、深度和空间
Width, depth and space
摘要：
树的宽度度量树的深度，也被称为顶点排序，中心着色和消除树的高度，是一个很好的概念，最近又重新引起了人们的兴趣。由于有界树深度的图比有界树或路径宽度的图更受限制，我们对这种额外结构的算法效用感兴趣。在消极的一面，我们表明，每一个动态规划算法的树深度分解深度~ $t$不能解决支配集与$O((3-ε)^t \cdot \log n)$空间的任何$ε> 0$。这一结果意味着相同的空间下界的动态规划算法的树和路径分解。我们通过证明3-着色的$O((3-ε)^t \cdot \log n)$和顶点覆盖的$O((2-ε)^t \cdot \log n)$的空间下界来补充这个结果。这形式化的共同直觉，图分解的动态规划算法必然会消耗大量的空间，并补充了已知的结果的时间复杂性的问题限制到低树宽类。然后，我们表明，树深度本身的分支算法的设计。这类算法通常比动态编程算法具有明显的优势：a）它们比基于动态编程的算法使用更少的空间，b）它们易于并行化，c）它们在终止之前提供可能的解决方案。具体来说，我们设计了一个纯分支算法，运行在时间$t^{O(t^2)}\cdot n$和使用空间$O(t^3 \log t + t \log n)$和分支和动态规划的混合，实现了运行时间的$O(3^t \log t \cdot n)$，同时使用$O(2^t t \log t + t \log n)$空间。为了完整起见，分别包括空间复杂度为$O(t \cdot \log n)$和时间复杂度为$O(3^t \cdot n)$和$O(2^t\cdot n)$的3-着色和顶点覆盖算法。
英文摘要：
The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. Since graphs of bounded treedepth are more restricted than graphs of bounded tree- or pathwidth, we are interested in the algorithmic utility of this additional structure. On the negative side, we show that every dynamic programming algorithm on treedepth decompositions of depth~$t$ cannot solve Dominating Set with $O((3-ε)^t \cdot \log n)$ space for any $ε> 0$. This result implies the same space lower bound for dynamic programming algorithms on tree and path decompositions. We supplement this result by showing a space lower bound of $O((3-ε)^t \cdot \log n)$ for 3-Coloring and $O((2-ε)^t \cdot \log n)$ for Vertex Cover. This formalizes the common intuition that dynamic programming algorithms on graph decompositions necessarily consume a lot of space and complements known results of the time-complexity of problems restricted to low-treewidth classes. We then show that treedepth lends itself to the design of branching algorithms. This class of algorithms has in general distinct advantages over dynamic programming algorithms: a) They use less space than algorithms based on dynamic programming, b) they are easy to parallelize and c) they provide possible solutions before terminating. Specifically, we design for Dominating Set a pure branching algorithm that runs in time $t^{O(t^2)}\cdot n$ and uses space $O(t^3 \log t + t \log n)$ and a hybrid of branching and dynamic programming that achieves a running time of $O(3^t \log t \cdot n)$ while using $O(2^t t \log t + t \log n)$ space. Algorithms for 3-Coloring and Vertex Cover with space complexity $O(t \cdot \log n)$ and time complexity $O(3^t \cdot n)$ and $O(2^t\cdot n)$, respectively, are included for completeness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
