A greedy approximation algorithm for the group Steiner problem

AbstractIn 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 the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265–285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73–91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59–63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an O((log∑i|gi|)1+ε·logm) approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of O(log|V|) in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(maxi|gi|)·logm) provided by the LP based approaches

Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Minimum Latency Submodular Cover

We study the Minimum Latency Submodular Cover problem (MLSC), which consists of a metric $(V,d)$ with source $r\in V$ and $m$ monotone submodular functions $f_1, f_2, ..., f_m: 2^V \rightarrow [0,1]$. The goal is to find a path originating at $r$ that minimizes the total cover time of all functions. This generalizes well-studied problems, such as Submodular Ranking [AzarG11] and Group Steiner Tree [GKR00]. We give a polynomial time O(\log \frac{1}{\eps} \cdot \log^{2+\delta} |V|)-approximation algorithm for MLSC, where $\epsilon>0$ is the smallest non-zero marginal increase of any $\{f_i\}_{i=1}^m$ and $\delta>0$ is any constant. We also consider the Latency Covering Steiner Tree problem (LCST), which is the special case of \mlsc where the $f_i$s are multi-coverage functions. This is a common generalization of the Latency Group Steiner Tree [GuptaNR10a,ChakrabartyS11] and Generalized Min-sum Set Cover [AzarGY09, BansalGK10] problems. We obtain an $O(\log^2|V|)$-approximation algorithm for LCST. Finally we study a natural stochastic extension of the Submodular Ranking problem, and obtain an adaptive algorithm with an O(\log 1/ \eps) approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [GoemansV06] and Shared Filter Evaluation [MunagalaSW07, LiuPRY08].Comment: 23 pages, 1 figur