Worst-case performance of Rayward-Smith's Steiner tree heuristic

Abstract. An integrative overview of the algorithmic characteristics of three well-known polynomialtime heuristics for the undirected Steiner minimum tree problem: shortest path heuristic (SPH), distance network heuristic (DNH), and average distance heuristic (ADH) is given. The performance of these single-pass heuristics (and some variants) is compared and contrasted with several heuristics based on repetitive applications of the SPH. It is shown that two of these repetitive SPH variants generate solutions that in general are better than solutions obtained by any single-pass heuristic. The worst-case time complexity of the two new variants is O(pn 3) and O(p3n2), while the worst-case time complexity of the SPH, DNH, and ADH is respectively 0(pn2), O(m + n log n), and O(n 3) where p is the number of vertices to be spanned, n is the total number of vertices, and m is the total number of edges. However, use of few simple tests is shown to provide large reductions of problem instances (both in terms of vertices and in term of edges). As a consequence, a substantial speed-up is obtained so that the repetitive variants are also competitive with respect to running times.

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“…Furthermore, as recently proved by Waxman and Imase [21], the worst-case error ratio is bounded by the same constant as for the DNH.…”

Section: (V)mentioning

confidence: 59%

“…Furthermore, it remains an open problem whether the worst-case error ratio for the K-SPH is bounded by 2. The close relation of the K-SPH to both SPH and DNH (see the next section) can perhaps make it possible to use worst-case error ratio proofs for these heuristics [20], [21] in connection with the K-SPH.…”

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confidence: 99%

Path-distance heuristics for the Steiner problem in undirected networks

Winter

Smith

1992

Algorithmica

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“…The standard 2-approximation algorithm for the Steiner tree problem discovered independently by [17,20] belongs to the class of FIXED priority greedy algorithms. In the restricted case when the edges of the graph have weights either 1 or 2, known as the STEINER(1,2) problem, Bern and Plassmann [6] proved that the average distance heuristic [21,24], is a 4 3 -approximation. The average distance heuristic, however, does not seem to fit our priority model.…”

Section: The Metric Steiner Tree Problemmentioning

confidence: 99%

Models of Greedy Algorithms for Graph Problems

Davis

Impagliazzo

2007

Algorithmica

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Borodin et al. (Algorithmica 37(4):295-326, 2003) gave a model of greedy-like algorithms for scheduling problems and Angelopoulos and Borodin (Algorithmica 40(4): 2004) extended their work to facility location and set cover problems. We generalize their model to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an abstract model that captures the intrinsic power and limitations of greedy algorithms for various graph optimization problems, as Borodin et al. (Algorithmica 37(4):295-326, 2003) did for scheduling. We prove bounds on the approximation ratio achievable by such algorithms for basic graph problems such as shortest path, weighted vertex cover, Steiner tree, and independent set. For example, we show that, for the shortest path problem, no algorithm in the FIXED priority model can achieve any approximation ratio (even one dependent on the graph size), but the well-known Dijkstra's algorithm is an optimal ADAPTIVE priority algorithm. We also prove that the approximation ratio for weighted vertex cover achievable by ADAPTIVE priority algorithms is exactly 2. Here, a new lower bound matches the known upper bounds (Johnson in J. Comput. Syst. Sci. 9(3): 1974). We give a number of other lower bounds for priority algorithms, as well as a new approximation algorithm for minimum Steiner tree problem with weights in the interval [1, 2].

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“…The Prim based heuristic [12], the Kruskal based heuristic [13,15], the distance graph heuristic [3,6,8,10], and the average distance heuristic [11,14] all achieve an approximation ratio of 2 Ϫ 2/͉R͉. We denote by T PH the tree obtained with the Prim based heuristic, by T KH the tree obtained with the Kruskal heuristic, by T DGH the tree obtained with the distance graph heuristic, and by T ADH the tree obtained with the average distance heuristic.…”

mentioning

confidence: 99%

The Push Tree problem

Havet

Wennink

2004

Networks

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In this article, we introduce the Push Tree problem, which exposes the tradeoffs between the use of push and pull mechanisms in information distribution systems. One of the interesting features of the Push Tree problem is that it provides a smooth transition between the minimum Steiner Tree and the Shortest Path problems. We present initial complexity results and analyze heuristics. Moreover, we discuss what lessons can be learned from the static and deterministic Push Tree problem for more realistic scenarios characterized by high uncertainty and changing information request and update patterns.

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Worst-case performance of Rayward-Smith's Steiner tree heuristic

Abstract: In this paper, we prove that the worst case performance of the Steiner tree approximation algorithm by Rayward-Smith is within two times optimal and that two is the best bound in the sense that there are instances for which RS will do worse than any value less than two.

Cited by 25 publications

References 9 publications

Path-distance heuristics for the Steiner problem in undirected networks

Path-distance heuristics for the Steiner problem in undirected networks

Models of Greedy Algorithms for Graph Problems

The Push Tree problem

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