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.
A well-studied special case of is the problem ( ). Here we are given an undirected weighted graph, a root vertex @math , and a collection of @math groups @math . The goal is to compute the cheapest tree that spans @math and at least one vertex from each group @math . The best-known polynomial-time approximation factor for is @math due to Garg et al. @cite_2 . Their algorithm uses probabilistic distance-based tree embeddings @cite_24 @cite_3 as a subroutine. Chekuri and Pal @cite_31 presented an @math approximation that runs in quasi-polynomial time. On the negative side, Halperin and Krauthgamer @cite_21 showed that admits no @math -approximation algorithm, for any constant @math , unless @math . This implies the same hardness for , hence for and .
@cite_21 @cite_3 @cite_24 @cite_2 @cite_31
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In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems. This paper provides a novel technique for the analysis of randomized algorithms for optimization problems on metric spaces, by relating the randomized performance ratio for any, metric space to the randomized performance ratio for a set of "simple" metric spaces. We define a notion of a set of metric spaces that probabilistically-approximates another metric space. We prove that any metric space can be probabilistically-approximated by hierarchically well-separated trees (HST) with a polylogarithmic distortion. These metric spaces are "simple" as being: (1) tree metrics; (2) natural for applying a divide-and-conquer algorithmic approach. The technique presented is of particular interest in the context of on-line computation. A large number of on-line algorithmic problems, including metrical task systems, server problems, distributed paging, and dynamic storage rearrangement are defined in terms of some metric space. Typically for these problems, there are linear lower bounds on the competitive ratio of deterministic algorithms. Although randomization against an oblivious adversary has the potential of overcoming these high ratios, very little progress has been made in the analysis. We demonstrate the use of our technique by obtaining substantially improved results for two different on-line problems. Given a weighted graph with some subsets of vertices called groups, the group Steiner tree problem is to find a minimum-weight subgraph which contains at least one vertex from each group. We give a randomized algorithm with a polylogarithmic approximation guarantee for the group Steiner tree problem. The previous best approximation guarantee was O(i2k1 i) in time O(nik2i) (Charikar, Chekuri, Goel, and Guha). Our algorithm also improves existing approximation results for network design problems with location-based constraints and for the symmetric generalized traveling salesman problem. Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting.