2-Approximation Algorithm for Finding a Spanning Tree with Maximum Number of Leaves

Solis-Oba, Roberto

doi:10.1007/3-540-68530-8_37

Cited by 89 publications

(56 citation statements)

References 8 publications

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“…Additionally, MLSTP was also shown to be MAX-SNP-hard (Galbiati et al 1994), implying that an > 0 exists such that it is NP-hard to solve the problem within an approximation factor of (1 + ). Factor of 3 approximation algorithms for MLSTP are suggested in Ravi (1992, 1998) while a factor of 2 algorithm could be found in Solis-Oba (1998).…”

Section: Introductionmentioning

confidence: 99%

Reformulations and solution algorithms for the maximum leaf spanning tree problem

2010

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Given a graph G = (V, E), the maximum leaf spanning tree problem (MLSTP) is to find a spanning tree of G with as many leaves as possible. The problem is easy to solve when G is complete. However, for the general case, when the graph is sparse, it is proven to be NP-hard. In this paper, two reformulations are proposed for the problem. The first one is a reinforced directed graph version of a formulation found in the literature. The second recasts the problem as a Steiner arborescence problem over an associated directed graph. Branch-and-Cut algorithms are implemented for these two reformulations. Additionally, we also implemented an improved version of a MLSTP Branch-and-Bound algorithm, suggested in the literature. All of these algorithms benefit from pre-processing tests and a heuristic suggested in this paper. Computational comparisons between the three algorithms indicate that the one associated with the first reformulation is the overall best. It was shown to be faster than 123 290 A. Lucena et al. the other two algorithms and is capable of solving much larger MLSTP instances than previously attempted in the literature.

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Section: Introductionmentioning

confidence: 99%

Reformulations and solution algorithms for the maximum leaf spanning tree problem

2010

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“…This way they reduced the running time to be almost linear while preserved the approximation factor of 3. The currently known best approximation factor is 2, and is achieved by Solis-Oba [7] whose algorithm is based on the local improvement idea in [6]. Finally we mention another degree-based spanning tree optimization problem which is also a generalization of the Hamiltonian Path problem.…”

Section: Introductionmentioning

confidence: 98%

Approximating the Maximum Internal Spanning Tree problem

Salamon

2009

Theoretical Computer Science

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a b s t r a c tGiven an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G. This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation -unless P = NP -, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem.In this paper, we improve this approximation ratio by giving an O(|V (G)| 4 )-time 7/4-approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal-dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G. For this problem we present a (2∆(G) − 3)-approximation algorithm, where ∆(G) is the maximum vertex-degree of G. A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O(|V (G)| 4 ) time for graphs with no pendant vertices.

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“…In Section 2 we give a linear time 2-approximation algorithm for the MAXIST problem based on depth first search. In Section 3 we show that a refined version of the depth first search algorithm provides a 3 2 -approximation on clawfree graphs (graphs not containing K 1,3 as an induced subgraph) and a 6 5 -approximation on cubic graphs (3-regular graphs). It is worth mentioning that Lu and Ravi [3,4] gave the first constant factor approximation for the problem of finding a spanning tree with a maximum number of leaves.…”

Section: Introduction and Notationsmentioning

confidence: 99%

“…Several of these problems have an objective function based on the degrees of nodes of the spanning tree (see for example [1,4,6]). This model is extremely useful when designing networks where the cost of devices to install depends fundamentally on the required routing functionality (ending, forwarding, or routing a connection).…”

Section: Introduction and Notationsmentioning

confidence: 99%