A Parameterized Approximation Algorithm for The Shallow-Light Steiner Tree Problem This project was supported by the Natural Science Foundation of Fujian Province (2012J05115), Doctoral Fund of Ministry of Education of China for Young Scholars (20123514120013) and Fuzhou University Development Fund (2012-XQ-26).

A Parameterized Approximation Algorithm for The Shallow-Light Steiner Tree Problem thanks: This project was supported by the Natural Science Foundation of Fujian Province (2012J05115), Doctoral Fund of Ministry of Education of China for Young Scholars (20123514120013) and Fuzhou University Development Fund (2012-XQ-26).

Longkun Guo, Kewen Liao School of Mathematics and Computer Science, Fuzhou University, China
School of Computer Science, University of Adelaide, Australia
   XiuJun Wang School of Mathematics and Computer Science, Fuzhou University, China
School of Computer Science, University of Adelaide, Australia
Abstract

For a given graph with a terminal set and a selected root , a positive integer cost and a delay on every edge and a delay constraint , the shallow-light Steiner tree (SLST) problem is to compute a minimum cost tree spanning the terminals of , in which the delay between root and every vertex is restrained by . This problem is NP-hard and very hard to approximate. According to known inapproximability results, this problem admits no approximation with ratio better than factor unless [10], while it admits no approximation ratio better than for unless [2]. Hence, the paper focus on parameterized algorithm for SLST. We firstly present an exact algorithm for SLST with time complexity , where and are the number of terminals and vertices respectively. This is a pseudo polynomial time parameterized algorithm with respect to the parameterization: “number of terminals”. Later, we improve this algorithm such that it runs in polynomial time , and computes a Steiner tree with delay bounded by and cost bounded by the cost of an optimum solution, where is any small real number. To the best of our knowledge, this is the first parameterized approximation algorithm for the SLST problem.

Keywords:
S

hallow light Steiner tree, parameterized approximation algorithm, directed Steiner tree, exact algorithm, auxiliary graph, pseudo-polynomial time complexity.

1 Introduction

The well-known shallow-light Steiner tree problem (or namely the delay restrained minimum Steiner tree problem) is defined as below:

Definition 1

For a graph with a terminal set , a root vertex , a cost function , a delay function , and a delay bound , the shallow-light Steiner tree (SLST) problem is to compute a minimum cost Steiner tree spanning all terminals of , such that the delay from to every terminal in is not larger than .

For notation briefness, we assume in graph , and use SLST and to denote the shallow-light Steiner tree problem and an optimal shallow-light Steiner tree respectively. For the SLST problem, bifactor approximation algorithms have been developed.

Definition 2

An algorithm is a bifactor -approximation for the SLST problem, if and only if for every instance of SLST, computes a Steiner tree in polynomial time, such that the delay from to every terminal in is bounded by and the cost of is bounded by times of the cost of the optimal solution.

Noting that single factor -approximation is identical to bifactor -approximation for SLST, we use them interchangeably in the text.

Related Work. It is known that the SLST problem is NP-hard, and can not be approximated better than factor unless [10]. This is because the group Steiner tree problem can be embedded into this problem. Furthermore, no polylogarithmic approximation within polynomial time complexity has been developed. The best work is a long standing result due to Charikar et al, which is a polylogarithmic approximation in quasi-polynomial time, i.e. factor- approximation within time complexity [3]. Due to the difficulty in single factor approximation algorithm design, bifactor approximation has been investigated. Hajiaghayi et al presented an -approximation algorithm that runs in polynomial time [8]. Besides, Kapoor and Sarwat gave an approximation with bifactor , where is an input parameter [9]. The last algorithm is an approximation that improves the cost of the tree, and is with bifactor when [9].

The SLST problem remains hard to approximate even when . In that case, this problem becomes the shallow light spanning tree (SLT) problem, which has broad applications in network design, VLSI and etc. For computational complexity, the SLT problem is claimed to be with inapproximability hardness of [12]. For approximation, Charikar et al’s ratio with time complexity [3] is still the best single factor result. Naor and Schieber gave an approximation bifactor of , i.e. with delay and cost bounded by 2 times and times of that of the optimal solution respectively [12]. To the best of our knowledge, these are the best long standing approximation ratios. Some special cases of the SLT problem are also interesting. If edge cost is equal to the delay for each edge, the SLT problem remains NP-hard and admit no approximation algorithms with bifactor for any and [11], while the best possible result for SLST is a -approximation [5]. Moreover, the SLT problem remains NP-hard when all edge delays are equal, but polynomially solvable when all edge costs are equal [13]. For the equal-delay case, namely the hop constrained minimum spanning tree problem, Althaus et al have presented an approximation with a ratio of in [1].

Another two important special cases of the SLST problem is when is constant or when all edge delays are equal. Unfortunately, for the former case, SLST can not be approximated better than a factor of for even unless [2], since the Set Cover problem can be embedded into this case. Bar-Ila et al also developed a factor- approximation for the cases of in the same paper, achieving the best possible ratio. When all edge delays are equal, namely the hop constrained minimum Steiner tree problem, it is open that if there exists factor- approximation for this problem, as the spanning case.

Our Contribution. The first result of this paper is an exact algorithm, with time complexity , for the SLST problem. This result indicates that if the number of terminal and the delay constraint are bounded, the SLST problem is polynomial solvable. Our technique is mainly based on constructing an auxiliary graph, where every Steiner tree satisfies the delay constraint, i.e. in the auxiliary graph, we only need to compute Steiner tree without considering the delay constraint. Though its time complexity seems terrible, the exact algorithm is efficient for real-world applications for , particularly when or all edge delays are equal (the hop constrained minimum Steiner tree problem).

On the theoretical side, we note that this algorithm runs in pseudo polynomial time (for constant ), since appears in the formula of the time complexity. The second result is to improve this time complexity to polynomial time following a similar line of polynomial-time approximation scheme (PTAS) design, such that it computes a Steiner tree with delay bounded by and cost bounded by the cost of an optimum solution.

2 A Parameterized Approximation Algorithm for the Shallow Light Steiner Problem

In this section we shall approximate the shallow-light Steiner tree (SLST) problem. Firstly and intuitively, our main observation is that the difficulty of computing a comes from obeying the given delay constraint. Therefore, our key idea is to construct an auxiliary directed graph where there exists only cost (i.e. no delay) on edges, such that every Steiner tree (spanning the same terminal set) in corresponds to a Steiner tree that satisfies the given delay constraint in . Secondly since the directed Steiner tree problem is known parameterized tractable with respect to the parameterization: “number of terminals”[6, 4], an exact algorithm is immediately obtained; then an approximation algorithm with ratio can be derived from the exact algorithm by a method of shrinking the value of . The approximation algorithm computes a with delay bounded by and cost bounded by the cost of an optimum .

2.1 Construction of the Auxiliary Graph

Though different in technique details, the key idea to construct the auxiliary graph is similar to the auxiliary graph used to balance the cost and delay of disjoint shortest paths in [7]: using layer graphs. For a given graph with positive integer cost and delay on every edge, and a delay constraint , the layer graph , i.e. the auxiliary graph to be constructed, contains vertices, terminals and edges roughly as in the following:

  1. vertices corresponding to every vertex ;

  2. edges corresponding to every edge and with ;

  3. one terminal , corresponding to every terminal , together with cost-0 edges that connect auxiliary vertices of to the auxiliary terminal;

Therefore, has vertices, edges, and terminals. The construction is formerly as in Algorithm 1 (An example of such construction is as depicted in Figure 1).

Input: Graph , a set of terminals , a root vertex , cost and delay on every edge , and a delay constraint ;

Output: Auxiliary acyclic graph and the terminal set therein, .

  1. , ;

  2. For each do

    1. ;

    2. If then

      1. , and set for each ;

      2. ;

  3. For each that do

    , and set for each ;

  4. For each do

    , and set .

  5. Return and .

Algorithm 1 Construction of auxiliary graph .
Figure 1: Construction of acyclic graphs: (a) is the original graph, in which are terminals; (b) is the constructed auxiliary graph, in which are terminals.

It remains to show that the -rooted minimum cost directed Steiner tree in corresponds to a -rooted minimum in .

Lemma 3

A minimum cost directed Steiner tree rooted at in contains at most one vertex of for each .

Proof

Let be a -rooted minimum cost directed Steiner tree in . Suppose contains and . Then we show that is not minimum and get a contradiction. Let be except removing the edge entering and replacing every edge in the subtree of that roots at , say by edge . Apparently, spanning the same terminal set as . That is, there exists a directed Steiner tree with less cost than in . This contradicts with the fact that is minimum.

Theorem 4

Let be the set of terminal vertices in . Then there exists a -rooted directed Steiner tree spanning of minimum cost in iff there exists a Steiner tree spanning of minimum cost with delay between and every terminal restrained by in .

Proof

Let be a minimum cost directed Steiner tree rooted at in . Let be a subgraph of , in which if and only if there exists . Then because , we have . It remains to show is a Steiner tree. From Lemma 3, holds for every . So a path connecting to a terminal in corresponds to a path connecting to a terminal in . Then since every terminal of is reachable from in , all terminals of are connected to in . Besides, because is a tree, contains no loops or parallel edges. Therefore, is a Steiner tree of .

Let be a Steiner tree in . Then there is a unique path from root to every other vertex of . Hence, every vertex of has a unique delay from . Let contains edge for every , and edge if and only if , where is the delay from to in and the delay from to . Since the delay of from to every vertex in is not larger than , edge belongs to , and hence . Then because every is reachable from and no loop or parallel edge exists following the construction of , is a Steiner tree in with cost . This completes the proof.

2.2 A parameterized Approximation Algorithm for Shallow-Light Steiner Tree

This subsection shall give an exact algorithm and a parameterized approximation algorithm for the SLST problem. From Theorem 4, an algorithm for the SLST problem can be obtained by computing a minimum cost directed Steiner tree in . Unfortunately, it is known that the (minimum) directed Steiner tree problem is NP-hard and maybe even more difficult to approximate than SLST, i.e. only a quasi-polynomial time algorithm with a polylogarithmic approximation factor has been developed[3]. However, when the number of the terminals is a constant, the directed Steiner tree problem is polynomial solvable, as stated in the proposition below:

Proposition 5

[6]An optimum solution to the directed Steiner tree problem can be computed within , where and are the number of terminals and vertices respectively.

Following Algorithm 1, Theorem 4 and Proposition 5, we could now state the exact algorithm for the SLST problem as in the following:

Input: Graph , , , cost function and delay function , a delay constraint , and auxiliary graph with ,

Output: , an optimum solution to the SLST problem.

  1. ;

  2. Compute a minimum cost Steiner tree in , say spanning the terminal of by the method of [4];

  3. For every do

    If then ;

  4. Return .

Algorithm 2 An exact algorithm for SLST.

Following Theorem 4 and Proposition 5, we immediately have the correctness of Algorithm 2. For time complexity, since contains edges, vertices and terminals, it takes time to compute a minimum Steiner tree in . Hence, we have:

Theorem 6

Algorithm 2 solved the SLST problem correctly, and runs in time .

We note that Algorithm 2 runs in pseudo-polynomial time, since the formula of the time complexity contains . However, following the technique of polynomial-time approximation scheme (PTAS) design, a parameterized approximation algorithm for the SLST problem could proceed as: firstly compute , which is except the delay of every edge is sat to , such that the value of delay constraint is shrunken from to a polynomial on ; secondly construct graph with the new delay on edges; and finally run Algorithm 2 on the auxiliary graph of the new delay. Formally, the parameterized approximation algorithm for the SLST problem is as in the following:

Input: A given parameter , graph , , , cost and delay on every edge , and a delay constraint ;

Output: , an approximation solution to the SLST problem.

  1. For every edge of do

    ;

    /* Compute .*/

  2. Construct auxiliary graph and compute using Algorithm 1;

  3. Compute a minimum cost Steiner tree subjected to the new delay constraint by applying Algorithm 2 on and with respect to the new delay;

  4. Return .

Algorithm 3 A parameterized approximation algorithm for SLST.

Following Algorithm 3, the delay constraint in is . Then from Lemma 6, the time complexity of the algorithm is after shrinking to . Hence, we have

Lemma 7

Algorithm 3 runs in time . .

It remains to show the approximation of the algorithm, which is given by the following theorem:

Theorem 8

Algorithm 3 computes a Steiner tree spanning all terminals of in with cost bounded by the cost of an optimum , and delay bounded by .

Proof

Clearly, an optimum in will satisfy the new delay constraint in . Then since is a optimum solution to SLST in , it is with cost not larger than the cost of an optimum in .

It remains to show the delay of in . Let be an arbitrary path in . Then since the delay of in is bounded by , we have:

(1)

Following the definition of , holds, and hence:

(2)

Combining Inequality (1) and (2) yields:

(3)

Therefore, following Inequality (3), the delay of in is:

This completes the proof.

3 Conclusion

This paper investigated exact algorithms and then parameterized approximation algorithms for the SLST problem. The first result is an exact algorithm that computes optimum in time , and the second result is a factor- approximation algorithm with time complexity . A problem remained open is whether design of algorithms for the SLST problem with polylogarithmic approximation ratio is possible.

References

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