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About 7,318 papers for “Dominating set”

Dominating Set

Ming‐Yang Kao · Encyclopedia of Algorithms · 2008

Cited by 0

Dominating Set Games

Velzen, S. van · RePEc: Research Papers in Economics

In this paper we study cooperative cost games arising from domination problems on graphs.We introduce three games to model the cost allocation problem and we derive a necessary and su cient condition for the balancedness of all three games.Furthermore we study concavity of these games.

Cited by 0

Distributed minimum dominating set . . .

Christoph Lenzen, Yvonne-Anne Pignolet, Roger Wattenhofer · 2013

A dominating set is a subset of the nodes of a graph such that all nodes are in the set or adjacent to a node in the set. A minimum dominating set approximation is a dominating set that is not much larger than a dominating set with the fewest possible number of nodes. This article summarizes the state-of-the-art with respect to finding minimum dominating set approximations in distributed systems, where each node locally executes a protocol on its own, communicating with its neighbors in order to achieve a solution with good global properties. Moreover, we present a number of recent results for specific families of graphs in detail. A unit disk graph is given by an embedding of the nodes in the Euclidean plane, where two nodes are joined by an edge exactly if they are in distance at most one. For this family of graphs, we prove an asymptotically tight lower bound on the trade-off between time complexity and approximation ratio of deterministic algorithms. Next, we consider graphs of small arboricity, whose edge sets can be decomposed into a small number of forests. We give two

Cited by 0

A Dominating Set Algorithm

Sang-Un Lee · Journal of the Korea Society of Computer and Information · 2013

본 논문은 아직까지 정확한 해를 다항시간으로 구하는 알고리즘이 존재하지 않아 NP-완전 문제로 알려진 지배집합 (DS) 문제의 정확한 해를 선형시간으로 구하는 알고리즘을 제안하였다. 제안된 알고리즘은 그래프의 간선이 존재하지 않을 때까지 최소차수 <TEX>${\delta}(G)$</TEX>를 가진 정점 u의 인접정점들 중 최대차수 <TEX>${\Delta}(G)$</TEX>를 가진 정점 v를 최소 독립지배집합(MIDS)의 원소로 포함시키고 v의 부속 간선을 삭제하는 방법을 반복적으로 수행하여 구하였다. MIDS로부터 최소 지배집합 (MDS)으로 변환시키고, MDS로부터 최소연결 DS (MCDS)로 변환시키는 방법으로 DS 관련 모든 문제의 정확한 해를 구할 수 있었다. 제안된 알고리즘을 10개의 다양한 그래프에 적용한 결과 정확한 해를 선형 시간복잡도 O(n)으로 구하는데 성공하였다. 결국, 제안된 지배집합 알고리즘은 지배집합 문제가 P-문제임을 증명하였다. This paper proposes a linear-time algorithm that has been designed to obtain an accurate solution for Dominating Set (DS) problem, which is known to be NP-complete due to the deficiency of polynomial-time algorithms that successfully derive an accurate solution to it. The proposed algorithm does so by repeatedly assigning vertex v with maximum degree <TEX>${\Delta}(G)$</TEX>among vertices adjacent to the vertex v with minimum degree <TEX>${\delta}(G)$</TEX> to Minimum Independent DS (MIDS) as its element and removing all the incident edges until no edges remain in the graph. This algorithm finally transforms MIDS into Minimum DS (MDS) and again into Minimum Connected DS (MCDS) so as to obtain the accurate solution to all DS-related problems. When applied to ten different graphs, it has successfully obtained accurate solutions with linear time complexity O(n). It has therefore proven that Dominating Set problem is rather a P-problem.

Cited by 0Open access

Dominating Set Knapsack: Profit Optimization on Dominating Sets

Sipra Singh · ArXiv.org · 2025

In a large-scale network, we want to choose some influential nodes to make a profit by paying some cost within a limited budget so that we do not have to spend more budget on some nodes adjacent to the chosen nodes; our problem is the graph-theoretic representation of it. We define our problem, Dominating Set Knapsack, by attaching the knapsack problem with the dominating set on graphs. Each vertex $v~(\in V) $ is associated with a cost factor $w(v)$ and a profit amount $α(v)$. We aim to choose some vertices within a fixed budget $(s)$ that give maximum profit so that we do not need to choose their 1-hop neighbors. We show that the Dominating Set Knapsack problem is strongly NPC even when restricted to bipartite graphs, but weakly NPC for star graphs. We present a pseudo-polynomial time algorithm for trees in time $O(n\cdot min\{s^2, (α(V))^2\})$. We show that Dominating Set Knapsack is unlikely to be Fixed Parameter Tractable (FPT) by proving that it is W[2]-hard parameterized by the solution size. We developed FPT algorithms with running time $O(4^{tw}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$ and $O(2^{vck-1}\cdot n^{O(1)} min\{s^2,{α(V)}^2\})$, where $tw$ represents the $tw$ of the given graph $G(V,E)$, $vck$ is the solution size of the Vertex Cover Knapsack, $s$ is the capacity or size of the knapsack and $α(V)=\sum_{v\in V}α(v)$. We obtained similar results for other variants $k-$Dominating Set Knapsack and Minimal Dominating Set Knapsack, where $k$ is the size of the dominating set.

Cited by 0Open access

Dominating Set Reconfiguration with Answer Set Programming

Masato Kato, Mutsunori Banbara, Torsten Schaub et al. · Theory and Practice of Logic Programming · 2024

Abstract The dominating set reconfiguration problem is defined as determining, for a given dominating set problem and two among its feasible solutions, whether one is reachable from the other via a sequence of feasible solutions subject to a certain adjacency relation. This problem is PSPACE-complete in general. The concept of the dominating set is known to be quite useful for analyzing wireless networks, social networks, and sensor networks. We develop an approach to solve the dominating set reconfiguration problem based on answer set programming (ASP). Our declarative approach relies on a high-level ASP encoding, and both the grounding and solving tasks are delegated to an ASP-based combinatorial reconfiguration solver. To evaluate the effectiveness of our approach, we conduct experiments on a newly created benchmark set.

Cited by 0Open access

Algorithm and Hardness results on Liar's Dominating Set and $k$-tuple Dominating Set

Sandip Banerjee, Sujoy Bhore · arXiv (Cornell University) · 2019

Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $D\subseteq V$ such that every vertex $u\in V\setminus D$ is adjacent to at least one vertex $v\in D$. That is, the set $D$ satisfies the condition that $|N[v]\cap D|\geq 1$ for each $v\in V$, where $N[v]$ is the closed neighborhood of $v$. In this paper, we study two variants of the classical dominating set problem: $\boldmath{k}$-tuple dominating set ($k$-DS) problem and Liar's dominating set (LDS) problem, and obtain several algorithmic and hardness results. On the algorithmic side, we present a constant factor ($\frac{11}{2}$)-approximation algorithm for the Liar's dominating set problem on unit disk graphs. Then, we obtain a PTAS for the $\boldmath{k}$-tuple dominating set problem on unit disk graphs. On the hardness side, we show a $Ω(n^2)$ bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar's dominating set problem as well as for the $\boldmath{k}$-tuple dominating set problem. Furthermore, we prove that the Liar's dominating set problem on bipartite graphs is W[2]-hard.

Cited by 0Open access

Positive Influence Dominating Set Games

Guangyuan Wang, Hua Wang, Xiaohui Tao et al. · 2014

Motivated by applications in social networks, a new type of dominating set named Positive Influence Dominating Set (PIDS) has been studied in the literature. In this paper, we investigate cooperative cost games arising from PIDS problem on social network graphs. We propose two new game models, Rigid PIDS Game and Relaxed PIDS Game, and focus on their cores. First, a relationship between the cores of both games is obtained. Next, we also prove that the core of the relaxed PIDS game is nonempty if and only if there is no integrality gap for the relaxation linear programming of the PIDS problem on graph G.

Cited by 2