Parallel cleaning of a network with brushes

Gaspers, Serge; Messinger, Margaret-Ellen; Nowakowski, Richard J.; Prałat, Paweł

doi:10.1016/j.dam.2009.11.003

Cited by 15 publications

(14 citation statements)

References 7 publications

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“…The cleaning model, introduced in [15,16], is a combination of the chip-firing game and edge searching on a simple finite graph. (See also [11] where the parallel version of the process is studied.) The brush number of a graph G defined below corresponds to the minimum total imbalance of G which is used in the graph drawing theory.…”

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confidence: 97%

Cleaning Regular Graphs with Brushes

Alon¹,

Prałat²,

Wormald³

2009

SIAM J. Discrete Math.

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A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean d-regular graphs in this model, focusing on the asymptotic number for random d-regular graphs. We use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method to find the (asymptotic) number of brushes needed to clean a random d-regular graph using this algorithm (for fixed d). We further show that for any d-regular graph on n vertices at most n(d + 1)/4 brushes suffice and prove that, for fixed large d, the minimum number of brushes needed to clean a random d-regular graph on n vertices is asymptotically almost surely n 4 (d + o(d)), thus solving a problem raised in [M.E. Messinger, R.J. Nowakowski, P. Pra lat, and N. Wormald, Cleaning random d-regular graphs with brushes using a degree-greedy algorithm, in Combinatorial and Algorithmic Aspects of Networking, Lecture Notes in Comput. Sci. 4852, Springer, BerlinHeidelberg, 2007, pp. 13-26].

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confidence: 97%

Cleaning Regular Graphs with Brushes

Alon¹,

Prałat²,

Wormald³

2009

SIAM J. Discrete Math.

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“…The minimum number of brushes required to clean a given graph G = (V, E) is denoted by b(G), and for a given integer k, the problem of determining whether b(G) ≤ k, is NP-complete [4]. Bounds on b(G) are discussed in [9], cleaning random graphs was considered in [1,8,11], whereas the parallel version of the model has been studied in [5]. Other variations on this problem have been studied in [6,10,12].…”

Section: Introductionmentioning

confidence: 99%

“…The Agent Clearing Path problem is a variant of the problem of cleaning a graph with brushes, see e.g. [1,4,5,9,8,11]. In this problem, given a connected graph, initially with all dirty vertices and edges, a number of agents, called brushes, are placed on some vertices of the graph.…”

Section: Introductionmentioning

confidence: 99%

Clearing Connections by Few Agents

Levcopoulos

Lingas

Nilsson

et al. 2014

Lecture Notes in Computer Science

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Abstract. We study the problem of clearing connections by agents placed at some vertices in a directed graph. The agents can move only along directed paths. The objective is to minimize the number of agents guaranteeing that any pair of vertices can be connected by a underlying undirected path that can be cleared by the agents. We provide several results on the hardness, approximability and parameterized complexity of the problem. In particular, we show it to be: NP-hard, 2-approximable in polynomial-time, and solvable exactly in O(αn 3 2 2α ) time, where α is the number of agents in the solution. In addition, we give a simple linear-time algorithm optimally solving the problem in digraphs whose underlying graphs are trees. Finally, we discuss a related problem, where the task is to clear with a minimum number of agents a subgraph of the underlying graph containing its spanning tree. We show that this problem also admits a 2-approximation in polynomial time.

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“…The problem has been introduced in [7] where some initial results can be found as well. Cleaning random graphs was considered in [1,8,10] whereas the parallel version of the model has been studied in [3].…”

Section: Introductionmentioning

confidence: 99%

Clean the graph before you draw it!

Gaspers

Messinger

Nowakowski

et al. 2009

Information Processing Letters

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We prove a relationship between the Cleaning problem and the Balanced VertexOrdering problem, namely that the minimum total imbalance of a graph equals twice the brush number of a graph. This equality has consequences for both problems. On one hand, it allows us to prove the N P-completeness of the Cleaning problem, which was conjectured by Messinger et al. [7]. On the other hand, it also enables us to design a faster algorithm for the Balanced Vertex-Ordering problem [6].

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Parallel cleaning of a network with brushes

Cited by 15 publications

References 7 publications

Cleaning Regular Graphs with Brushes

Cleaning Regular Graphs with Brushes

Clearing Connections by Few Agents

Clean the graph before you draw it!

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