Simeon Kublenz, Sebastian Siebertz, Alexandre Vigny
We show that the dominating set problem admits a constant factor approximation in a constant number of rounds in the LOCAL model of distributed computing on graph classes with bounded expansion. This generalizes a result of Czygrinow et al. for graphs with excluded topological minors.
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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.
Tight Bounds for Constant-Round Domination on Graphs of High Girth and Low Expansion
本文研究了高周长低扩展图中的$k$跳支配问题，在端口号模型和本地算法中分别给出了紧致的常数轮近似比上界与下界，证明了常数时间近似比可达到次线性。
Domination in graphs with bounded propagation: algorithms, formulations and hardness results
在\textsc{Dominating Set}问题和\textsc{Power Dominating Set}（PDS）问题之间引入了一个问题层次，称为$\ell$ -轮幂支配集（$\ell$ -轮PDS，简称PDS）问题.对于$\ell=1$，这是\textsc{Dominating Set}问题，对于$\ell\geq n-1$，这是PDS问题;这里$n$表示输入图中的节点数。在PDS中，目标是找到功率支配所有节点的最小大小的节点集合$S$，其中节点$v$是功率支配的，如果（1）$v$在$S$中或者它在$S$中具有邻居，或者（2）$v$具有邻居$u$，使得$u$及其除了$v$之外的所有邻居都是功率支配的。注意，规则（1）与用于\textsc{Dominating Set}问题的规则相同，并且规则（2）是迭代地应用的一类传播规则。$\ell$轮PDS问题具有与PDS相同的规则集，除了我们在至多$\ell-1$轮中“并行”地应用规则（2）。证明了在一般图中，即使对$\ell=4$，$\ell$ -圆PDS也不能比$2^{\log^{1-ε}{n}}$更好地逼近.我们提供了一个动态规划算法，以解决$\ell$轮PDS最佳在多项式时间内有界树宽的图。我们提出了一个PTAS（多项式时间近似方案）$\ell$ -圆PDS平面图$\ell=O(\tfrac{\log{n}}{\log{\l…
Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) has at least $d(v)$ neighbors in $S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the $k$-tuple dominating set problem (this $k$ is different from the solution size), and so on, and its approximability and inapproximability …
We show that the dominating set problem admits a constant factor approximation in a constant number of rounds in the LOCAL model of distributed computing on graph classes with bounded expansion. This generalizes a result of Czygrinow et al. for graphs with excluded topological minors to very general classes of uniformly sparse graphs. We demonstrate how our general algorithm can be modified and fine-tuned to compute an ($11+ε$)-approximation (for any $ε>0)$ of a minimum dominating set on planar graphs. This improves on the previously best known approximation factor of 52 on planar graphs, whic…
In m-eternal domination attacker and defender play on a graph. Initially, the defender places guards on vertices. In each round, the attacker chooses a vertex to attack. Then, the defender can move each guard to a neighboring vertex and must move a guard to the attacked vertex. The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper, we study the m-eternal domination number of cactus graphs. We consider two variants of the m-eternal domination number: one allows multiple guards to occupy a single vertex, the second variant…
We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\…
A set $D\subseteq V$ is called a $k$-tuple dominating set of a graph $G=(V,E)$ if $\left| N_G[v] \cap D \right| \geq k$ for all $v \in V$, where $N_G[v]$ denotes the closed neighborhood of $v$. A set $D \subseteq V$ is called a liar's dominating set of a graph $G=(V,E)$ if (i) $\left| N_G[v] \cap D \right| \geq 2$ for all $v\in V$ and (ii) for every pair of distinct vertices $u, v\in V$, $\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3$. Given a graph $G$, the decision versions of $k$-Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a $k$-tuple dom…
The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixed surface with bounded number of crossings on each edge. The algorithm is based on the following approximate min-max characterization. A subset A of vertices of a graph G is d-independent if the distance…
Near-Optimal Distributed Domination in Planar Graphs
本文提出平面图上最小支配集的确定性$(8+\varepsilon)$-近似算法，在LOCAL模型常数轮内完成，通过新的结构界将近似比从$11+\varepsilon$降至接近最优下界$7$。
We investigate the parameterized complexity of generalisations and variations of the dominating set problem on classes of graphs that are nowhere dense. In particular, we show that the distance-d dominating-set problem, also known as the (k,d)-centres problem, is fixed-parameter tractable on any class that is nowhere dense and closed under induced subgraphs. This generalises known results about the dominating set problem on H-minor free classes, classes with locally excluded minors and classes of graphs of bounded expansion. A key feature of our proof is that it is based simply on the fact tha…
On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width
本文针对有界树宽图的混合支配问题，提出基于幂值分配与动态规划的新算法，实现固定参数可解，时间复杂度为O(3^{tw^2}×tw^2×|V|)，并改进至O(6^{tw}×|V|)。
The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with l leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We extend our approach to obtain similar results for Connected Do…
We unify and extend previous kernelization techniques in sparse classes [6,17] by defining water lilies and show how they can be used in bounded expansion classes to construct linear bikernels for (r, c)-Dominating Set, (r, c)-Scattered Set, Total r-Domination, r-Roman Domination, and a problem we call (r, [λ, μ])-Domination (implying a bikernel for r-Perfect Code). At the cost of slightly changing the output graph class our bikernels can be turned into kernels. We further demonstrate how these constructions can be combined to create 'multikernels', meaning graphs that represent kernels for mu…
From FPT to W[P]: Classifying Zero Forcing, Power Domination and Their Variants
本研究分类了零强制与电力支配问题变体的参数化复杂性，发现初始化、轮数限制和强制阈值的微小变化会导致从FPT到W[P]-完全的复杂性剧变。
Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication. In this paper we provide algorithms that are simultaneously round- and message-optimal for a number of well-studied distributed…
Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r + 1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the 'residual' set R := V (G) \ N [X]. Our work aims to answer two questions: How does the…
Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances
研究有界树宽图上距离d的独立集和支配集问题的连续版本，提出δ-分散和δ-覆盖问题的算法，并证明其最优性。
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…
Constant-factor approximation of maximum distance-2 independent set in graphs of bounded merge-width
针对有界半径-2合并宽度图，提出基于线性规划的常数因子近似算法求解最大距离-2独立集问题，并证明该结果与最小支配集问题的近似算法等价，且半径-2合并宽度界是紧的。
Parameterized Complexity of Domination Problems Using Restricted Modular Partitions
本文通过引入$\mathcal{G}$-模块基数参数，研究支配问题的参数化复杂性，证明解表是否可简洁表示决定了问题在模块基数下是FPT还是W[1]-困难，并据此获得多类支配问题在团宽度、模块宽度等参数下的新正负结果。
Residue Domination in Bounded-Treewidth Graphs
本文研究模剩余类约束下的顶点选择问题，提出树宽参数化算法，将运行时间改进为 m^{tw}·n^{O(1)}，并给出基于 SETH 的匹配下界。
The Massive Parallel Computation (MPC) model is a theoretical framework for popular parallel and distributed platforms such as MapReduce, Hadoop, or Spark. We consider the task of computing a large matching or small vertex cover in this model when the space per machine is $n^δ$ for $δ\in (0,1)$, where $n$ is the number of vertices in the input graph. A direct simulation of classic PRAM and distributed algorithms from the 1980s results in algorithms that require at least a logarithmic number of MPC rounds. We give the first algorithm that breaks this logarithmic barrier and runs in $\tilde O(\s…
Near-Optimal Distributed Ruling Sets for Trees and High-Girth Graphs
本文提出在树和高环长图中寻找裁定集的近最优分布式算法，改进了现有算法并降低了裁定距离。
We provide a deterministic CONGEST algorithm to constant factor approximate the minimum dominating set on graphs of bounded arboricity in $O(\log n)$ rounds. This improves over the well-known randomized algorithm of Lenzen and Wattenhofer[DISC2010] by making it a deterministic algorithm.
Capacitated Domination generalizes the classic Dominating Set problem by specifying for each vertex a required demand and an available capacity for covering demand in its closed neighborhood. The objective is to find a minimum-sized set of vertices that can cover all of the graph's demand without exceeding any of the capacities. In this paper we look specifically at domination with hard-capacities, where the capacity and cost of a vertex can contribute to the solution at most once. Previous complexity results suggest that this problem cannot be solved (or even closely approximated) efficiently…
We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt {log k}) in the case of the Arora-Rao-Vazirani relaxation, and wi…
Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP
研究针对每个变量最多出现在 d 个约束中的布尔 MAX 2-CSP 实例，通过阈值舍入方案实现 $\widetilde{\Omega}(1/d^4)$ 改进，MAX 2-SAT 有更优比率算法，推广了 MAX CUT 算法，还表明相关问题有界度实例也有类似改进。
We present deterministic distributed algorithms for computing approximate maximum cardinality matchings and approximate maximum weight matchings. Our algorithm for the unweighted case computes a matching whose size is at least $(1-\eps)$ times the optimal in $Δ^{O(1/\eps)} + O\left(\frac{1}{\eps^2}\right) \cdot\log^*(n)$ rounds where $n$ is the number of vertices in the graph and $Δ$ is the maximum degree. Our algorithm for the edge-weighted case computes a matching whose weight is at least $(1-\eps)$ times the optimal in $\log(\min\{1/\wmin,n/\eps\})^{O(1/\eps)}\cdot(Δ^{O(1/\eps)}+\log^*(n))$…
Domination and Coverage Problems under Vulnerability Constraints
研究脆弱性约束下的支配与覆盖问题，定义\(k\)-顶点最大支配率等问题，基于无预算变体提出近似算法，为\(k\)-Max DRVV问题提供\(O(k/n)\)近似，还对带脆弱顶点的支配集等问题进行研究并给出近似算法。
We consider the problem of finding a minimum edge cost subgraph of a graph satisfying both given node-connectivity requirements and degree upper bounds on nodes. We present an iterative rounding algorithm of the biset LP relaxation for this problem. For directed graphs and $k$-out-connectivity requirements from a root, our algorithm computes a solution that is a 2-approximation on the cost, and the degree of each node $v$ in the solution is at most $2b(v) + O(k)$ where $b(v)$ is the degree upper bound on $v$. For undirected graphs and element-connectivity requirements with maximum connectivity…
Approximate Turing kernelization and lower bounds for domination problems
本文研究支配集等问题的近似图灵核化，证明以树宽为参数时它们不存在常数因子近似多项式图灵核化，但在树宽与最大度联合参数化下可获得(1+ε)-近似多项式图灵核化。
Distributed Dominating Set With Optimal Rounds and Message Size in Bounded Arboricity Graphs
针对树状度α的图，提出一种确定性分布式算法，在最优轮复杂度O(logΔ/loglogΔ)和最优消息大小1比特下，实现O(α logΔ/loglogΔ)近似，无需预先知道α。
Dynamic domination and independence in sparse graphs
研究稀疏图中动态支配与独立问题，给出动态数据结构，能维护特定查询答案，随机且有错误概率界定，给出摊销更新时间。\(r = 1\)时可改进结构，还能维护支配集最小大小近似，有相应更新时间。
Classic Round-Up Variant of Fast Unsigned Division by Constants: Algorithm and Full Proof
针对常数整数除法开销问题，本文研究快速无符号常数除法的经典向上取整变体，给出完整正确性与可行性证明，并提出面向有界被除数的简化变体。
In this note, we describe a $α_{GW} + \tildeΩ(1/d^2)$-factor approximation algorithm for Max-Cut on weighted graphs of degree $\leq d$. Here, $α_{GW}\approx 0.878$ is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg and Florén. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.
Locally Checkable Labeling (LCL) problems include essentially all the classic problems of $\mathsf{LOCAL}$ distributed algorithms. In a recent enlightening revelation, Chang and Pettie [arXiv 1704.06297] showed that any LCL (on bounded degree graphs) that has an $o(\log n)$-round randomized algorithm can be solved in $T_{LLL}(n)$ rounds, which is the randomized complexity of solving (a relaxed variant of) the Lovász Local Lemma (LLL) on bounded degree $n$-node graphs. Currently, the best known upper bound on $T_{LLL}(n)$ is $O(\log n)$, by Chung, Pettie, and Su [PODC'14], while the best known …
We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+ε)$ approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate $B$ such that $B \le B^* \le \lceil (1+ε)B \rceil +1$ where $B^*$ is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribut…
Treewidth is a measure of how tree-like a graph is. It has many important algorithmic applications because many NP-hard problems on general graphs become tractable when restricted to graphs of bounded treewidth. Algorithms for problems on graphs of bounded treewidth mostly are dynamic programming algorithms using the structure of a tree decomposition of the graph. The bottleneck in the worst-case run time of these algorithms often is the computations for the so called join nodes in the associated nice tree decomposition. In this paper, we review two different approaches that have appeared in t…
Motivated by applications in sociology, economy and medicine, we study variants of the Target Set Selection problem, first proposed by Kempe, Kleinberg and Tardos. In our scenario one is given a graph $G=(V,E)$, integer values $t(v)$ for each vertex $v$ (\emph{thresholds}), and the objective is to determine a small set of vertices (\emph{target set}) that activates a given number (or a given subset) of vertices of $G$ \emph{within} a prescribed number of rounds. The activation process in $G$ proceeds as follows: initially, at round 0, all vertices in the target set are activated; subsequently …
Graph burning is a simple model for the spread of social influence in networks. The objective is to measure how quickly a fire (e.g., a piece of fake news) can be spread in a network. The burning process takes place in discrete rounds. In each round, a new fire breaks out at a selected vertex and burns it. Meanwhile, the old fires extend to their neighbours and burn them. A burning schedule selects where the new fire breaks out in each round, and the burning problem asks for a schedule that burns all vertices in a minimum number of rounds, termed the burning number of the graph. The burning pr…
Stochastic Multi-round Submodular Optimization with Budget
研究带预算的随机多轮次次模最大化（SBMSm）问题，提出多项式时间动态规划算法和0.316近似贪心算法，并引入预算自适应间隙证明其介于1.582和2之间。
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