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When Diameter Matters: Parameterized Approximation Algorithms for Bounded Diameter Minimum Steiner Tree Problem

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Abstract

Given a graph G with a set of terminals, two weight functions c and d defined on the edge set of G, and a bound D, a popular NP-hard problem in designing networks is to find the minimum cost Steiner tree (under function c) in G, to connect all terminals in such a way that its diameter (under function d) is bounded by D. Marathe et al. (J. Algoritm. 28(1), 142–171, 1998) proposed an (O(lnn),O(lnn)) approximation algorithm for this bicriteria problem, where n is the number of terminals. The first factor reflects the approximation ratio on the diameter bound D, and the second factor indicates the cost-approximation ratio. Later, Kapoor and Sarwat (Theory Comput. Syst. 41(4), 779–794, 2007) introduced a parameterized approximation algorithm with performance guarantee of \((O(p \cdot \frac {\ln n}{\ln p}), O(\frac {\ln n}{\ln p}))\) for any input value p>1, by which one can improve the approximation factor for cost at the price of worsening the approximation factor of diameter. In this paper, we consider the reverse scenario in which minimizing the diameter of the solution is more important. We first propose a parameterized approximation algorithm with performance guarantee of \((O(\frac {\ln n}{\ln p}),O(p \cdot H_{p} \cdot \frac {\ln n}{\ln p}))\), where H p = 1+1/2+…+1/p is the p th harmonic number. Parameter p is part of the input and this algorithm runs in polynomial time for constant values of p. We also present another algorithm with approximation ratio of \((O(\frac {\ln n}{\ln p}),O(\mu \cdot p \cdot \frac {\ln n}{\ln p}))\) which relies on the approximation factor (μ) of the NP-hard problem min-degree constrained minimum spanning tree.

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Notes

  1. In order to do this, we use the definition introduced by Howell [23] as follows. Assume that we have function \(f : \mathbb {N}^{k} \rightarrow \mathbb {R}^{\geq 0}\), and \(\hat {f}(n_{1}, \ldots , n_{k}) = \max \{f(i_{1}, \ldots , i_{k}) | 0\leq i_{j} \leq n_{j}, 1 \leq j \leq k \}\). Now, g(n 1,…,n k ) = O(f(n 1,…,n k )) if and only if there exist constants \(c_{0} \in \mathbb {N}\) and \(c_{1} \in \mathbb {R}^{>0}\) such that for all n 1,…,n k ≥c 0, we have g(n 1,…,n k ) ≤ c 1 f(n 1,…,n k ) and \(\hat {g}(n_{1},{\ldots } ,n_{k}) \leq c_{1} \hat {f}(n_{1},{\ldots } ,n_{k})\).

  2. In the Set Cover Problem we are given a set U of m elements and a set S of n subsets of U where the union of these n subsets is equal to U. The problem asks for a minimum number of subsets whose union becomes U (we say U has been covered). In the weighted version, each subset is assigned a positive weight and the problem asks for the collection of subsets with minimum total weight that covers U.

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  1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Ali Mashreghi & Alireza Zarei

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  1. Ali Mashreghi
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  2. Alireza Zarei
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Correspondence to Ali Mashreghi.

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Mashreghi, A., Zarei, A. When Diameter Matters: Parameterized Approximation Algorithms for Bounded Diameter Minimum Steiner Tree Problem. Theory Comput Syst 58, 287–303 (2016). https://doi.org/10.1007/s00224-015-9615-7

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