A tight local algorithm for the minimum dominating set problem in outerplanar graphs

Distributed distance domination in graphs with no $K_{2,t}$-minor

We prove that a simple distributed algorithm finds a constant approximation of an optimal distance-$k$ dominating set in graphs with no $K_{2,t}$-minor. The algorithm runs in a constant number of rounds. We further show how this procedure can be used to give a distributed algorithm which given $ε>0$ and $k,t\in \mathbb{Z}^+$ finds in a graph $G=(V,E)$ with no $K_{2,t}$-minor a distance-$k$ dominating set of size at most $(1+ε)$ of the optimum. The algorithm runs in $O(\log^*{|V|})$ rounds in the Local model. In particular, both algorithms work in outerplanar graphs.

Distributed Distance-$r$ Dominating Set on Sparse High-Girth Graphs

The dominating set problem and its generalization, the distance-$r$ dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance-$r$ independent set problem. By result of Kuhn et al. we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with hig…

Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs

We describe a simple deterministic $O( \varepsilon^{-1} \log Δ)$ round distributed algorithm for $(2α+1)(1 + \varepsilon)$ approximation of minimum weighted dominating set on graphs with arboricity at most $α$. Here $Δ$ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation [Kuhn, Moscibroda, and Wattenhofer JACM'16]. Our algorithm improves on all the previous results (that work only for unweighted graphs) including a r…

局部逼近的下界

Lower Bounds for Local Approximation

在确定性分布式算法的研究中，通常假设每个节点具有唯一的$O(\log n)$位标识符。我们证明，对于一般类的图问题，本地算法（恒定时间分布式算法）不需要这样的标识符：端口编号和方向是足够的。我们的结果适用于所谓的简单PO-可检查图优化问题，这包括许多经典的包装和覆盖问题，如顶点覆盖，边缘覆盖，匹配，独立集，控制集，和边缘控制集。我们专注于有界度图的情况下，并表明，如果一个本地算法发现一个简单的PO-可检查图的问题的帮助下，唯一标识符的常数因子近似，然后可以实现匿名网络上相同的近似比。作为我们结果的推论，并通过以前的工作，我们得到了一个紧的下限的局部逼近的最小边支配集问题。我们的主要技术工具是一个代数结构的均匀有序图：我们说，一个图是$(α,r)$ -齐，如果它的节点是线性有序的，使一个$α$分数的节点有成对同构半径- $r$邻域。证明了对任意$α< 1$和任意$r$、$k$和$g$，存在围长至少为$g$的有限$(α,r)$ -齐次$2k$ -正则图.

局部性质的局部认证：紧界、权衡与新参数

Local certification of local properties: tight bounds, trade-offs and new parameters

本文研究局部认证中证书大小不依赖网络规模而依赖其他参数的性质，为k-可着色性、距离t支配集和完美匹配证明了紧界，并揭示了如平面图完美匹配仅需两比特等意外结果。

局部计算：下限和上限

Local Computation: Lower and Upper Bounds

什么东西可以被计算，以及计算效率如何，是计算机科学的核心问题。毫不奇怪，在分布式系统和网络研究中，一个同样基本的问题是什么可以用\emph{distributed}方式计算。更准确地说，如果网络的节点必须仅基于其局部邻域中的信息来做出决策，那么它们能够计算或近似全局（优化）问题吗？本文给出了最小顶点覆盖、最小（连通）支配集、最大匹配、最大独立集和最大匹配等优化问题局部计算的第一个多对数下界.此外，我们提出了一个新的分布式算法求解一般覆盖和包装线性规划。对于某些问题，该算法是紧的下限，对其他人来说，它是一个分布式的近似计划。总之，我们的下限和上限建立了一大类问题的局部可计算性和可逼近性，表征了解决这些任务需要多少局部信息。