Probabilistic graph-coloring in bipartite and split graphs

Bourgeois, Nicolas; Croce, Federico Della; Escoffier, Bruno; Murat, Cécile; Paschos, Vangélis Th.

doi:10.1007/s10878-007-9112-2

Cited by 7 publications

(6 citation statements)

References 21 publications

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“…This path can be transformed into an elementary Hamiltonian path T by using suitable shortcuts as shown in Fig. 11b where these shortcuts are the boldfaced lines (edges) (5, 6), (8,9), (9, 10), (10,11), (12,13) and the whole path T is simply the path (i, i + 1), i = 1, . .…”

Section: General Metric Casementioning

confidence: 99%

“…In [6], the analysis of the probabilistic minimum travelling salesman problem, originally performed in [2,16], has been revisited and refined. Several other combinatorial problems have been handled in the probabilistic combinatorial optimization framework, like minimum coloring [8,26], maximum independent set and minimum vertex cover [24,25], longest path [23], Steiner tree problems [27,28]. Note also that probabilistic minimum spanning tree has also studied by [4] but under a very different probabilistic model.…”

Section: Introductionmentioning

confidence: 99%

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On the probabilistic min spanning tree Problem

2011

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We study a probabilistic optimization model for min spanning tree, where any vertex v i of the input-graph G(V, E) has some presence probability p i in the final instance G ⊂ G that will effectively be optimized. Suppose that when this "real" instance G becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modif ication strategy, that modifies the anticipatory tree T in order to fit G . The goal is to compute an anticipatory spanning tree of G such that, its modification for any G ⊆ G is optimal for G . This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively.

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Section: General Metric Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the probabilistic min spanning tree Problem

2011

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“…In the same spirit of the analysis of Christofides' algorithm ( [12]), consider now a left-to-right dfs ordering of the vertices ofT , the non-elementary path produced by this dfs traversal ofT and observe that in this traversal any edge is encountered at most twice. This path can be transformed into an elementary Hamiltonian path T ′ by using suitable shortcuts as shown in Figure 11(b) where these shortcuts are the boldfaced lines (edges) (5, 6), (8,9), (9, 10), (10,11), (12,13) and the whole path T ′ is simply the path (i, i + 1), i = 1, . .…”

Section: The Approximation Of Probabilistic Metric Min Spanning Treementioning

confidence: 99%

“…In [6], the analysis of the probabilistic minimum travelling salesman problem, originally performed in [2,17], has been revisited and refined. Several other combinatorial problems have been recently treated in the probabilistic combinatorial optimization framework, including minimum coloring ( [27,8]), maximum independent set and minimum vertex cover ( [25,26]), longest path ( [24]), Steiner tree problems ( [28,29]). Note also that probabilistic minimum spanning tree has also studied by [4] but under a very different probabilistic model.…”

Section: Introductionmentioning

confidence: 99%

On the PROBABILISTIC MIN SPANNING TREE problem

Boria

Murat

Paschos

2010

Proceedings of the International Multiconference on Computer Science and Information Technology

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We study a probabilistic optimization model for min spanning tree, where any vertex v i of the input-graph G(V, E) has some presence probability p i in the final instance G ′ ⊂ G that will effectively be optimized. Suppose that when this "real" instance G ′ becomes known, a spanning tree T , called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G ′. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G ′ ⊆ G is optimal for G ′. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively.

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“…Later, this approach was extended to other problems such as the probabilistic vehicle routing problem [3], the probabilistic spanning tree problem [4]. Studies on this probabilistic approach continued in many others domains such as the probabilistic maximum independent set problem [5], [6], the probabilistic longest path problem [7], the probabilistic minimum vertex covering problem [8], the probabilistic minimum coloring problem [9], the probabilistic graph-coloring in bipartite problem [10] and the probabilistic steiner tree problem [11]. The probabilistic approach has been extended on the combinatorial problems not defined on graph such probabilistic bin packing problem [12], [13] and the probabilistic scheduling problem [14].…”

Section: Introductionmentioning

confidence: 99%

A probabilistic traveling salesman problem: a survey

Henchiri¹,

Bellalouna

Khaznaji

2014

Annals of Computer Science and Information Systems

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The Probabilistic Traveling Salesman Problem (PTSP) is a variation of the classic Traveling Salesman Problem (TSP) in which only a subset of potential nodes needs to be visited on any given instance of the problem. The number of nodes to be visited each time is a random variable. The objective is to find an a priori tour which minimizes the expected length,with the strategy of visiting the present nodes in a particular instance in the same order as they appear in the a priori tour. In this paper, we survey a number of results obtained for PTSP and we present the different approaches used for solving it.

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Probabilistic graph-coloring in bipartite and split graphs

Cited by 7 publications

References 21 publications

On the probabilistic min spanning tree Problem

On the probabilistic min spanning tree Problem

On the PROBABILISTIC MIN SPANNING TREE problem

A probabilistic traveling salesman problem: a survey

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