k-Steiner Tree Research Articles

On the restricted k-Steiner tree problem

On the restricted k-Steiner tree problem

Maximal Steiner Trees in the Stochastic Mean-Field Model of Distance

Consider the complete graph on n vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as log n/n, whereas the diameter (maximum distance between any two vertices) scales as 3 log n/n. Bollobás, Gamarnik, Riordan and Sudakov showed that, for any fixed k, the weight of the Steiner tree connecting k typical vertices scales as (k − 1)log n/n, which recovers Janson's result for k = 2. We extend this to show that the worst case k-Steiner tree, over all choices of k vertices, has weight scaling as (2k − 1)log n/n and finally, we generalize this result to Steiner trees with a mixture of typical and worst case vertices.

On Fixed Cost k-Flow Problems

In the Fixed Cost k-Flow problem, we are given a graph G = (V, E) with edge-capacities {ue?e ? E} and edge-costs {ce?e ? E}, source-sink pair s, t ? V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. By an approximation-preserving reduction from Group Steiner Tree problem to Fixed Cost k-Flow, we obtain the first polylogarithmic lower bound for the problem; this also implies the first non-constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k-Flow. In the Bipartite Fixed-Cost k-Flow problem, we are given a bipartite graph G = (A ? B, E) and an integer k > 0. The goal is to find a node subset S ⊆ A ? B of minimum size |S| such G has k pairwise edge-disjoint paths between S ? A and S ? B. We give an O(klogk)$O(\sqrt {k\log k})$ approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = |A| + |B|. In the Generalized-P2P problem we are given an undirected graph G = (V, E) with edge-costs and integer charges {bv : v ? V}. The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [11]. Besides that, it generalizes many problems such as Steiner Forest, k-Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log3+??n approximation scheme for it using Group Steiner Tree techniques.

Improved approximations for buy-at-bulk and shallow-light $$k$$ k -Steiner trees and $$(k,2)$$ ( k , 2 ) -subgraph

In this paper we give improved approximation algorithms for some network design problems. In the bounded-diameter or shallow-light $$k$$k-Steiner tree problem (SL$$k$$kST), we are given an undirected graph $$G=(V,E)$$G=(V,E) with terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, a cost function $$c:E\rightarrow \mathbb {R}^+$$c:E?R+, a length function $$\ell :E\rightarrow \mathbb {R}^+$$l:E?R+, a bound $$L>0$$L>0 and an integer $$k\ge 1$$k?1. The goal is to find a minimum $$c$$c-cost $$r$$r-rooted Steiner tree containing at least $$k$$k terminals whose diameter under $$\ell $$l metric is at most $$L$$L. The input to the buy-at-bulk $$k$$k-Steiner tree problem (BB$$k$$kST) is similar: graph $$G=(V,E)$$G=(V,E), terminals $$T\subseteq V$$T⊆V containing a root $$r\in T$$r?T, cost and length functions $$c,\ell :E\rightarrow \mathbb {R}^+$$c,l:E?R+, and an integer $$k\ge 1$$k?1. The goal is to find a minimum total cost $$r$$r-rooted Steiner tree $$H$$H containing at least $$k$$k terminals, where the cost of each edge $$e$$e is $$c(e)+\ell (e)\cdot f(e)$$c(e)+l(e)·f(e) where $$f(e)$$f(e) denotes the number of terminals whose path to root in $$H$$H contains edge $$e$$e. We present a bicriteria $$(O(\log ^2 n),O(\log n))$$(O(log2n),O(logn))-approximation for SL$$k$$kST: the algorithm finds a $$k$$k-Steiner tree with cost at most $$O(\log ^2 n\cdot \text{ opt }^*)$$O(log2n·opt?) where $$\text{ opt }^*$$opt? is the cost of an LP relaxation of the problem and diameter at most $$O(L\cdot \log n)$$O(L·logn). This improves on the algorithm of Hajiaghayi et al. (2009) (APPROX'06/Algorithmica'09) which had ratio $$(O(\log ^4 n), O(\log ^2 n))$$(O(log4n),O(log2n)). Using this, we obtain an $$O(\log ^3 n)$$O(log3n)-approximation for BB$$k$$kST, which improves upon the $$O(\log ^4 n)$$O(log4n)-approximation of Hajiaghayi et al. (2009). We also consider the problem of finding a minimum cost $$2$$2-edge-connected subgraph with at least $$k$$k vertices, which is introduced as the $$(k,2)$$(k,2)-subgraph problem in Lau et al. (2009) (STOC'07/SICOMP09). This generalizes some well-studied classical problems such as the $$k$$k-MST and the minimum cost $$2$$2-edge-connected subgraph problems. We give an $$O(\log n)$$O(logn)-approximation algorithm for this problem which improves upon the $$O(\log ^2 n)$$O(log2n)-approximation algorithm of Lau et al. (2009).

On the approximability of Dense Steiner Problems

We study the approximation complexity of the @e-Dense Steiner Tree Problem which was introduced by Karpinski and Zelikovsky (1998) [13]. They proved that for each @e>0, this problem admits a PTAS. Based on their method we consider here dense versions of various Steiner Tree problems. In particular, we give polynomial time approximation schemes for the @e-Dense k-Steiner Tree Problem, the @e-Dense Prize Collecting Steiner Tree Problem and the @e-Dense Group Steiner Tree Problem. We also show that the @e-Dense Steiner Forest Problem is approximable within ratio 1+O((@?ilog|Si|)/(@?i|Si|)) where S1,...,Sn are the terminal sets of the given instance. This ratio becomes small when the number of terminal sets is small compared to the total number of terminals.

Two-stage Robust Network Design with Exponential Scenarios

We study two-stage robust variants of combinatorial optimization problems on undirected graphs, like Steiner tree, Steiner forest, and uncapacitated facility location. Robust optimization problems, previously studied by Dhamdhere et al. (Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005), Golovin et al. (Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006), and Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007), are two-stage planning problems in which the requirements are revealed after some decisions are taken in Stage 1. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust k-Steiner tree problem, for example, one buys some edges in Stage 1. Then k terminals are revealed in Stage 2 and one has to buy more edges, at a higher cost, to complete the Stage 1 solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for k-Steiner tree), or any subset of k terminal-pairs (for k-Steiner forest), or any subset of k clients (for facility location). Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007) give an LP-based general framework for approximation algorithms for a class of two stage robust problems. Their framework cannot be used for network design problems like k-Steiner tree (see later elaboration). Their framework can be used for the robust facility location problem, but gives only a logarithmic approximation. We present the first constant-factor approximation algorithms for the robust k-Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate nearby vertices. For the robust k-Steiner forest problem on trees and with uniform multiplicative increase factor for Stage 2 (also known as inflation), we present a constant approximation. We show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question of (Dhamdhere et al. in Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005) and (Golovin et al. in Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006).

Approximating Buy-at-Bulk and Shallow-Light k-Steiner Trees

We study two related network design problems with two cost functions. In the buy-at-bulk k-Steiner tree problem we are given a graph G(V,E) with a set of terminals T⊆V including a particular vertex s called the root, and an integer k≤|T|. There are two cost functions on the edges of G, a buy cost b:E→ℝ+ and a distance cost r:E→ℝ+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑e∈H b(e)+∑t∈T−s dist(t,s) is minimized, where dist(t,s) is the distance from t to s in H with respect to the r cost. We present an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. The second and closely related one is bicriteria approximation algorithm for Shallow-light k-Steiner trees. In the shallow-light k-Steiner tree problem we are given a graph G with edge costs b(e) and distance costs r(e), and an integer k. Our goal is to find a minimum cost (under b-cost) k-Steiner tree such that the diameter under r-cost is at most some given bound D. We develop an (O(log n),O(log 3 n))-approximation algorithm for a relaxed version of Shallow-light k-Steiner tree where the solution has at least $\frac{k}{8}$terminals. Using this we obtain an (O(log 2 n),O(log 4 n))-approximation algorithm for the shallow-light k-Steiner tree and an O(log 4 n)-approximation algorithm for the buy-at-bulk k-Steiner tree problem. Our results are recently used to give the first polylogarithmic approximation algorithm for the non-uniform multicommodity buy-at-bulk problem (Chekuri, C., et al. in Proceedings of 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 677–686, 2006).

Approximate k -MSTs and k -Steiner trees via the primal-dual method and Lagrangean relaxation

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning k vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and k-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together with the primal-dual method for approximation algorithms. We also derive a constant factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.

An approximation algorithm for a bottleneck k-Steiner tree problem in the Euclidean plane

We study a bottleneck Steiner tree problem: given a set P={ p 1, p 2,…, p n } of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the problem.