Improved Approximation Algorithms for the Quality of Service Multicast Tree Problem

Karpiński, Marek; Măndoiu, Ion; Оlshevsky, А.G.; Zelikovsky, Alexander

doi:10.1007/s00453-004-1133-y

Cited by 12 publications

(11 citation statements)

References 17 publications

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“…Recall that an improvement for ℓ = 3 was posed as an open problem by Karpinski et al [13]. Also, for each of the cases 4 ≤ ℓ ≤ 100 our results in Theorem 2.8 improve the approximation ratios of eρ ≈ 2.718ρ and 2.454ρ guaranteed by Charikar et al [6] and by Karpinski et al [13], respectively. The graph of the approximation ratio of the composite algorithm (see Figure 5) for ℓ = 1, .…”

Section: Composite Algorithmsupporting

confidence: 53%

“…. , 100 (black curve), compared to the ratio t = eρ (red dashed line) guaranteed by the algorithm of Charikar et al[6] and t = 2.454ρ (green dashed line) guaranteed by the algorithm of Karpinski et al[13]. The table to the right lists the exact values for the ratio t/ρ.…”

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confidence: 98%

“…Instead of taking the union of the Steiner trees on each rounded level, Karpinski et al [13] contract them into the source in each step, which yields a 2.454ρ-approximation. They also gave a (1.265 + ε)ρ-approximation for the 2-level case.…”

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confidence: 99%

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Multi-level Steiner Trees

Ahmed

Angelini

Sahneh

et al. 2019

ACM J. Exp. Algorithmics

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In the classical Steiner tree problem, given an undirected, connected graph G = (V , E) with non-negative edge costs and a set of terminals T ⊆ V , the objective is to find a minimum-cost tree E ′ ⊆ E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ = ln(4) + ε < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T ℓ ⊂ · · · ⊂ T 1 ⊆ V , compute nested trees E ℓ ⊆ · · · ⊆ E 1 ⊆ E that span the corresponding terminal sets with minimum total cost.The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two simple O(ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ = 100 levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.Let G = (V , E) be an undirected, connected graph with positive edge costs c : E → R + , and let T ⊆ V be a set of vertices called terminals. A Steiner tree is a tree in G that spans T . The network (graph) Steiner tree problem (ST) is to find a minimum-cost Steiner tree E ′ ⊆ E, where the cost of E ′ is c(E ′ ) = e ∈E ′ c(e). ST is one of Karp's initial NP-hard problems [12]; see also a survey [21], an online compendium [11], and a textbook [18].Due to its practical importance in many domains, there is a long history of exact and approximation algorithms for the problem. The classical 2-approximation algorithm for ST [10] uses the metric closure of G, i.e., the complete edge-weighted graph G * with vertex set T in which, for every edge uv, the cost of uv equals the length of a shortest u-v path in G. A minimum spanning tree of G * corresponds to a 2-approximate Steiner tree in G.Currently, the last in a long list of improvements is the LP-based approximation algorithm of Byrka et al. [5], which has a ratio of ln(4) + ε < 1.39. Their algorithm uses a new iterative randomized rounding technique. Note that ST is APX-hard [4]; more concretely, it is NP-hard to approximate the problem within a factor of 96/95 [7]. This is in contrast to the geometric variant of the problem, where terminals correspond to points in the Euclidean or rectilinear plane. Both variants admit polynom...

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Section: Composite Algorithmsupporting

confidence: 53%

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confidence: 98%

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Multi-level Steiner Trees

Ahmed

Angelini

Sahneh

et al. 2019

ACM J. Exp. Algorithmics

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“…Somewhat closer to our needs are Steiner trees with bounded depth that have also been studied extensively in the literature, e.g. [21,23]. Constructing an efficient multicast tree under various constraints with regard to latency has been readily studied, e.g.…”

Section: Related Workmentioning

confidence: 97%

Maximizing total upload in latency-sensitive P2P applications

Douceur

Lorch

Moscibroda

2007

Proceedings of the Nineteenth Annual ACM Symposium on Parallel Algorithms and Architectures

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Motivated by an application in distributed gaming, we define and study the latency-constrained total upload maximization problem. In this problem, a peer-to-peer overlay network is modeled as a complete graph and each node vi has an upload bandwidth capacity ci and a set of receivers R(i). Each sender-receiver pair (vi, vj ), where vj ∈ R(i), is a request that should be satisfied, i.e., vi should send a data packet to each vj ∈ R(i). The goal is to find a set of at most n multicast-trees Ti of depth at most 2, such that each node can be part of multiple trees, all capacity constraints are met, and the number of satisfied requests is maximized. In this paper, we prove that the problem is NP-complete, and we present an algorithm with approximation ratio 1 − 2/ √ cmin, where cmin is the minimum upload capacity. Finally, we also study the impact of network coding on the quality and approximability of the solution.

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“…GOSST 문제 에서 중요하게 고려되는 서비스 등급의 수를 2와 3으로 제한 한 휴리스틱 알고리즘을 제안한 연구가 있었는데 [3], 이 연구 에서는 branch-and-bound 알고리즘에서 매우 큰 문제 영역 에서 좋은 결과를 얻을 수 있는 k-optimal 휴리스틱 알고리즘 도 제시하였다. M. Karpinski등은 GOSST를 멀티미디어 통신 의 멀티캐스팅에 적용한 개선된 휴리스틱을 발표하였다 [13].…”

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SGOSST Mechanism for Quality of Service In Network

Kim¹

2011

Journal of the Korea Society of Computer and Information

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Because of boost of communications devices furnishing diverse services and rapid expansion of mobile business, good use and management of the existing network system become very important. Also, offering service corresponding with user communication requirement grades which vary widely in each person, is vital for communication service provider. In this paper, SGOSST, a mechanism of efficient network construction with minimum cost for network QoS is proposed. In experiments, though spending 252. 97% more execution times, our SGOSST QoS network consumed 5.11% less connecting costs than the network constructed by weighted minimum spanning tree method. Therefore our mechanism can work well for efficient operation and service providing in the network formed with users and communication devices of various service requirement grade as smart/mobile equipment.

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Improved Approximation Algorithms for the Quality of Service Multicast Tree Problem

Cited by 12 publications

References 17 publications

Multi-level Steiner Trees

Multi-level Steiner Trees

Maximizing total upload in latency-sensitive P2P applications

SGOSST Mechanism for Quality of Service In Network

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