关于“连通集覆盖问题和容错连通集覆盖问题的算法”的注记

Constant approximation for fault-tolerant median problems via iterative rounding

In this paper, we study the fault-tolerant matroid median and fault-tolerant knapsack median problems. These two problems generalize many fundamental clustering and facility location problems, such as uniform fault-tolerant $k$-median, uniform fault-tolerant facility location, matroid median, knapsack median, etc. We present a versatile iterative rounding framework and obtain a unifying constant-factor approximation algorithm.

Fault-Tolerant Edge-Disjoint Paths -- Beyond Uniform Faults

The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform fault model. Such a model assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ may fail. While this approach yields problems with clean combinatorial structure and good algorithms, it often fails to capture the true nature of the scenario set coming from applications. One natural refinement of the uniform model is obtained by partitioning the set of resources into vulnerable and safe resources. The scenario set contains every subset of at most $k$ faulty …

Optimal Vertex Fault-Tolerant Spanners in Polynomial Time

Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex faults, and there are examples of input graphs on which this bound cannot be improved. However, these proofs work by analyzing the output spanner of a certain exponential-time greedy algorithm. In this work, we give the first algorithm that produces vertex fault tolerant spanners of optimal size and which runs in polynomial time. Specifically, we give a randomized…

Fault-Tolerant Distance Labeling for Planar Graphs

In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph $G$ such that from the labels of any three vertices $u,v,f$ we can infer the $u$-to-$v$ distance in the graph $G\setminus \{f\}$. We show that any directed weighted planar graph (and in fact any graph in a graph family with $O(\sqrt{n})$-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size $O(n^{2/3})$. We extend these labels in a way that allows us to also count the number of shortest paths, and provide additional upper and lower bounds for labels and oracles for …

容错k-中位数的常数因子近似算法

A Constant Factor Approximation Algorithm for Fault-Tolerant k-Median

本文研究了容错$k$ -median问题，给出了它的\emph{first}常数因子近似算法，在经典$k$ -median问题的容错推广中，每个客户端$j$需要被分配到至少$r_j \ge 1$不同的开放设施上. $j$的服务成本是其到$r_j$设施的距离之和，并且$k$ -中值约束将开放设施的数量限制为最多$k$。以前，只有在所有$r_j$都相同的特殊情况下才知道常数因子，而在一般情况下才知道对数近似比。此外，我们提出了第一个多项式时间算法的容错$k$ -中位数问题的路径或HST通过显示相应的LP总是有一个积分最优解。我们还考虑了容错设施的位置问题，$j$的服务成本可以是它的距离的加权和的$r_j$设施。我们给出了一个简单的常数因子近似算法，推广了以前的几个结果，只适用于非增权向量。

On Fault Tolerant Feedback Vertex Set

The study of fault-tolerant data structures for various network design problems is a prominent area of research in computer science. Likewise, the study of NP-Complete problems lies at the heart of computer science with numerous results in algorithms and complexity. In this paper we raise the question of computing fault tolerant solutions to NP-Complete problems; that is computing a solution that can survive the "failure" of a few constituent elements. This notion has appeared in a variety of theoretical and practical settings such as estimating network reliability, kernelization (aka instance…