A note on the Covering Steiner Problem (original) (raw)
Approximation algorithms for directed Steiner problems
Proceedings of the …, 1998
We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ...
A greedy approximation algorithm for the group Steiner problem
Discrete Applied Mathematics, 2006
In the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices {gi}i=1m. Each subset gi is called a group and the vertices in ⋃igi are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when
On subexponential running times for approximating directed Steiner tree and related problems
arXiv (Cornell University), 2018
This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1 − α) ln n, for a given parameter 0 < α < 1. What is the best possible running time for achieving such approximation ratio? This question was answered implicitly in the work of Moshkovitz [Theory of Computing, 2015]: Assuming both the Projection Games Conjecture (PGC) and the Exponential-Time Hypothesis (ETH), any ((1 − α) ln n)-approximation algorithm for Set-Cover must run in time at least 2 n c•α , for some small constant 0 < c < 1. We study the questions along this line. Our first contribution is in strengthening the above result. We show that under ETH and PGC the running time requires for any ((1 − α) ln n)-approximation algorithm for Set-Cover is essentially 2 n α. This (almost) settles the question since our lower bound matches the best known running time of 2 O(n α) for approximating Set-Cover to within a factor (1 − α) ln n given by Cygan et al. [IPL, 2009]. Our result is tight up to the constant multiplying the n α terms in the exponent. The lower bound of Set-Cover applies to all of its generalization, e.g., Group-Steiner-Tree, Directed-Steiner-Tree, Covering-Steiner-Tree and Connected-Polymatroid. We show that, surprisingly, in almost exponential running time, these problems reduce to Set-Cover. Specifically, we complement our lower bound by presenting an (1−α) ln n approximation algorithm for all aforementioned problems that runs in time 2 n α •log n • poly(m). We further study the approximation ratio in the regime of log 2−δ n for Group-Steiner-Tree and Covering-Steiner-Tree. Chekuri and Pal [FOCS, 2005] showed that Group-Steiner-Tree admits (log 2−α n)-approximation in time exp(2 log α+o(1) n), for any parameter 0 < α < 1. We show the running time lower bound of Group-Steiner-Tree: any (log 2−α n)-approximation algorithm for Group-Steiner-Tree must run in time at least exp((1 + o(1))log α−ǫ n), for any constant ǫ > 0, unless the ETH is false. Our result follows by analyzing the hardness construction of Group-Steiner-Tree due to the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for Covering-Steiner-Tree.
An algorithm f o r solving the Steiner problem on a f i n i t e undirected graph i s presented. s e t of graph arcs of minimum t o t a l length needed t o connect a specified s e t of k graph nodes. I f the entire graph contains n nodes, the algorithm requires time proportional t o ThiB algorithm computes the 3 The t h e requirement above includes the term n / 2 , which can be eliminated i f the s e t of shortest paths connecting each pair of nodes in the graph i s available. Also, the RAND Corporation, which through the U.S. A i r Force under t h e Project RAND, supported the authors during t h e i r i n i t i a l research on t h i s problem. i n p a r t i s permitted for any purposes of the United S t a t e s Government.
Generalized Steiner problems and other variants
2000
In this paper, we examine combinatorial optimization problems by considering the case where the set N (the ground set of elements) is expressed as a union of a finite number of m nonempty distinct subsets N 1 , . . . , N m . The term we use is the generalized Steiner problems coined after the Generalized Traveling Salesman Problem. We have collected a short list of classical combinatorial optimization problems and we have recast each of these problems in this broader framework in an attempt to identify a linkage between these "generalized" problems. In the literature one finds generalized problems such as the Generalized Minimum Spanning Tree (GMST), Generalized Traveling Salesman Problem (GTSP) and Subset Bin-packing (SBP). Casting these problems into the new problem setting has important implications in terms of the time effort required to compute an optimal solution or a "good" solution to a problem. We examine questions like "is the GTSP "harder" than the TSP?" for a number of paradigmatic problems starting with "easy" problems such as the Minimal Spanning Tree, Assignment Problem, Chinese Postman, Two-machine Flow Shop, and followed by "hard" problems such as the Bin-packing, and the TSP.
2-Approximation Algorithm for the Minimum Weighted Steiner Tree Problem
The Minimum Spanning Tree problem is well-known and has been studied extensively. The solution to this problem spans all vertices of a graph. Nonetheless, a more generalized problem-the Steiner Minimal Tree problem-is yet to be delved into thoroughly. The solution to this problem spans only a required subset of vertices of a graph. In this paper, we describe several algorithms to solve the Steiner Minimal Tree problem, and investigate specifically how the the Steiner Minimal Tree problem can be solved using a 2-approximation algorithm, with an application in essentially large instances. Due to the NP-hardness of the problem, approximation algorithms prove to be the sole feasible solution, unless the given instances are exceptionally small. We also discuss data structures that optimize the approximation algorithm for large graphs, and evaluate the time complexity of our implementation of the algorithm.
Algorithmica, 2012
We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of elementdisjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k elementconnected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns k 2 − 1 element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching 2.
An improved LP-based approximation for steiner tree
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010
The Steiner tree problem is one of the most fundamental AEÈ-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from ¾ to the current best ½ [Robins,Zelikovsky-SIDMA'05]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than ¾ [Vazirani,Rajagopalan-SODA'99].
The Minimum Degree Group Steiner Tree problem on Bounded Treewidth Graphs and Applications
2019
This paper studies two network design problems whose goals are to find a tree that minimizes the maximum degree. First is the Minimum Degree Group Steiner Tree problem (MD-GST), where we are given an n-vertex undirected graph G(V,E), and a collection of p subsets of vertices called groups {gi}i∈[p], and the goal is find a tree that contains at least one vertex from every group gi while minimizing the maximum degree. Second is the Minimum Degree k-Steiner Tree problem (MDkT), where we are given an n-vertex undirected graph G(V,E), a set of p terminals and a number k < p, and the goal is to find a tree that spans at least k terminals while minimizing the maximum degree. We study these two problems when an input graph has bounded treewidth. We present an O(log n/ log log n) approximation algorithm for MDGST on bounded treewidth graphs and showed that the latter problem, MDkT, can be reduced to MD-GST via a blackbox reduction that loses only an extra O(logn) factor in the approximati...