改进的有向图可达单调回路深度上界

Reachability in High Treewidth Graphs

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time although the space bound is $O(\log^2 n)$. Here, we study space efficient algorithms for deciding reachability that runs simultaneously in polynomial time. In this paper, we show that given an $n$ vertex directed graph of treewidth $w$ along with its tree decomposition, there exists …

小深度证明系统

Small Depth Proof Systems

语言L的证明系统是函数f，使得Range（f）恰好是L。在本文中，我们期待证明系统从电路复杂性的角度来看，研究证明系统的计算非常有限。我们研究的限制是：它们可以通过具有恒定深度的有界扇入回路（NC ^0）或具有O（log log n）深度但具有O（1）交替的有界扇入回路（polylog AC ^0）来计算。每个输出位依赖于非常少的输入位;因此这样的证明系统对应于一种定理证明对上的局部纠错。我们准确地确定了证明系统捕获所有正则语言所需的能力。我们证明了所有的正则语言都有多对数AC^0证明系统，并且从之前的结果（Beyersdorff et al，MFCS 2011，其中首次引入了NC^0证明系统）来看，这是紧密的。我们的技术还表明，MAJ具有polylog AC^0证明系统。我们探讨了TAUT是否有NC^0证明系统的问题。解决这个关于2 TAUT的问题，由于2 TAUT与图中的可达性密切相关，我们对可达性提出了同样的问题。我们证明了无向可达性和有向不可达性都有NC^0证明系统，但有向可达性仍然是开放的。在NP语言的证明系统需要多大的能力的背景下，我们观察到相当一部分NP语言的证明系统不需要AC^0的全部能力;它们有SAC^0或coSAC^0证明系统。

Reachability and Distances under Multiple Changes

Recently it was shown that the transitive closure of a directed graph can be updated using first-order formulas after insertions and deletions of single edges in the dynamic descriptive complexity framework by Dong, Su, and Topor, and Patnaik and Immerman. In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes. We show that the former problem can be maintained in DynFO$(+, \times)$ under changes affecting O($\frac{\log n}{\log \log n}$) nodes, for graphs with $…

三终端保持可达性的最小节点割：平面困难性与一般图的 \\(O(\sqrt n)\\)-近似

Three-Terminal Reachability-Preserving Minimum Node Cut: Planar Hardness and a General-Graph \(O(\sqrt n)\)-Approximation

研究三终端保持可达性的最小节点割问题，证明加权平面版本是NP完全的，并给出一般图的多项式时间\\(O(\sqrt n)\\)-近似算法。

Space-Efficient Algorithms for Reachability in Geometric Graphs

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem. For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and $O(m^{1/2}\log n)$ …

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

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。

Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on res…

圈长有界的强连通有向图

On Strongly Connected Digraphs with Bounded Cycle Length

MEG（最小等价图）问题是给定一个有向图，寻找一个小的边子集，使节点之间保持所有的可达关系。这个问题是NP难的。本文证明了对于每个有向圈最多有三条边的图，MEG问题等价于最大二部匹配问题，因而在多项式时间内是可解的。这导致改进了以前最好的近似算法的性能保证的一般问题“近似最小等效有向图”（1995年）。

动态图问题的超对数下界

Super-Logarithmic Lower Bounds for Dynamic Graph Problems

本文首次证明了动态有向无环图可达性查询在边插入下的超对数无条件下界，并提出了新的数据结构下界证明技术。