k-树中路径和匹配的对数空间算法
Log-space Algorithms for Paths and Matchings in k-trees
摘要：
可达性和最短路问题是一般图的NL-完全问题。对于树宽为2的图，已知它们在L中[JT 07]。然而，对于树宽度大于2的图，没有比NL更好的界限。在本文中，我们改进了k-树的这些界，其中k是一个常数。特别是，我们的论文的主要成果是日志空间算法的可达性在有向k-树，并计算最短和最长的路径在有向无环k-树。除了上面提到的路径问题之外，我们还考虑了决定k树是否具有完美匹配（决策版本）的问题，如果是，则寻找完美匹配（搜索版本），并证明这两个问题是L-完全的。已知这些问题存在于P和RNC中的一般图，以及SPL中的平面二分图[DKR 08]。我们的结果解决了这些问题的k-树类的复杂性。结果也适用于有界树宽图，当树分解作为输入。我们算法的核心技术是在对数空间中仔细实现分治方法，以及[JT 07]和[LMR 07]的一些想法。
英文摘要：
Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].
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