In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an @math -vertex weighted directed graph, a root @math , and a set of @math terminals @math , find a min-cost subgraph @math that has two edge vertex disjoint paths from @math to any @math . 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by , [SODA'09; JCSS] and has then been studied by [SODA'12; TALG] and Laekhanukit SODA'14]. However, no positive result was known except for the special case of a @math -shallow instance [Laekhanukit, ICALP'16]. We present an @math approximation algorithm for 2-DST that runs in time @math , for any @math . This implies a polynomial-time @math approximation for any constant @math , and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is @math -hard to approximate [Halperin and Krauthgamer, STOC'03]. As a by product, we obtain an algorithm with the same approximation guarantee for the @math -Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are @math -edge vertex connected.
As already mentioned, survivable network design is well studied in undirected (weighted) graphs. First, consider the edge connectivity version. The earliest work is initiated in early 80's by Frederickson and J ' a J ' a @cite_22 , where the authors studied the 2-Edge Connected Subgraph problem in both directed and undirected graphs. In the most general form of the problem, also known as the problem, we are given non-negative integer requirements @math for all pairs of vertices @math , and the goal is to find a min-cost subgraph that has @math edge-disjoint paths between @math and @math . Jain @cite_0 devised a @math -approximation algorithm for this problem. We remark that @math is the best known approximation factor even for @math @cite_39 , which is known as the problem. The classical problem is a special case of Steiner forest where all pairs share one vertex. Here the best known approximation factor is @math due to the work of Byrka et al. @cite_27 .
@cite_0 @cite_27 @cite_22 @cite_39
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We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math . In the survivable network design problem (SNDP), given an undirected graph and values r sub ij for each pair of vertices i and j, we attempt to find a minimum-cost subgraph such that there are r sub ij disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. K. (1999) propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELC-SNDP, or ELC). These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 (K. Jain, 2001), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O (log k) approximation algorithm, where k=max sub i,j r sub ij . VC-SNDP is not known to have a non-trivial approximation algorithm; however, recently L. Fleischer (2001) has shown how to extend the technique of K. Jain ( 2001) to give a 2-approximation algorithm in the case that r sub ij spl isin 0, 1, 2 . She also shows that the same techniques will not work for VC-SNDP for more general values of r sub ij . The authors show that these techniques can be extended to a 2-approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures. We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. In the survivable network design problem (SNDP), the goal is to find a minimum-cost spanning subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertex-disjoint paths connecting them. We give the first strong lower bound on the approximability of SNDP, showing that the problem admits no efficient @math ratio approximation for any fixed @math , unless @math . We show hardness of approximation results for some important special cases of SNDP, and we exhibit the first lower bound on the approximability of the related classical NP-hard problem of augmenting the connectivity of a graph using edges from a given set.