The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? and considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".
Approximation algorithms for robust optimization was initiated by @cite_31 : they study the case when the scenarios were explicitly listed, and gave constant-factor approximations for Steiner tree and facility location, and logarithmic approximations to mincut multicut problems. @cite_13 improved the mincut result to a constant factor approximation, and also gave an @math -approximation for robust shortest-paths. The algorithms in @cite_13 were also thresholded algorithms'' and the algorithms in this paper can be seen as natural extensions of that idea to more complex uncertainty sets and larger class of problems (the uncertainty set in @cite_13 only contained singleton demands).
@cite_31 @cite_13
1579892319 2167240010 2120899253 2949832116
Demand-robust versions of common optimization problems were recently introduced by [4] motivated by the worst-case considerations of two-stage stochastic optimization models. We study the demand robust min-cut and shortest path problems, and exploit the nature of the robust objective to give improved approximation factors. Specifically, we give a @math approximation for robust min-cut and a 7.1 approximation for robust shortest path. Previously, the best approximation factors were O(log n) for robust min-cut and 16 for robust shortest paths, both due to [4]. Our main technique can be summarized as follows: We investigate each of the second stage scenarios individually, checking if it can be independently serviced in the second stage within an acceptable cost (namely, a guess of the optimal second stage costs). For the costly scenarios that cannot be serviced in this way (“rainy days”), we show that they can be fully taken care of in a near-optimal first stage solution (i.e., by ”paying today”). We also consider “hitting-set” extensions of the robust min-cut and shortest path problems and show that our techniques can be combined with algorithms for Steiner multicut and group Steiner tree problems to give similar approximation guarantees for the hitting-set versions of robust min-cut and shortest path problems respectively. We study two-stage robustvariants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by [1], [6], and [4], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust Steiner tree problem, for example, one buys some edges in stage one after which some terminals are revealed. In the second stage, one has to buy more edges, at a higher cost, to complete the stage one 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 manyscenarios given implicitly. A scenario consists of any subset of kterminals (for Steiner tree), or any subset of kterminal-pairs (for Steiner forest), or any subset of kclients (for facility location). We present the first constant-factor approximation algorithms for the robust Steiner tree and robust uncapacitated facility location problems. For the robust Steiner forest problem with uniform inflation, we present an O(logn)-approximation and show that the problem with two inflation factors is impossible to approximate within O(log1 2 i¾? i¾?n) factor, for any constant i¾?> 0, unless NP has randomized quasi-polynomial time algorithms. Finally, we show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question by [1] and [6]. In the Steiner Forest problem, we are given terminal pairs si, ti, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi gave a primal-dual constant-factor approximation algorithm for this problem. Until this work, the only constant-factor approximations we know are via linear programming relaxations. In this paper, we consider the following greedy algorithm: Given terminal pairs in a metric space, a terminal is active if its distance to its partner is non-zero. Pick the two closest active terminals (say si, tj), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. It has long been open to analyze this greedy algorithm. Our main result shows that this algorithm is a constant-factor approximation. We use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest. In this paper we study the Steiner tree problem over a dynamic set of terminals. We consider the model where we are given an @math -vertex graph @math with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning a terminal set @math , which changes over time. The changes applied to the terminal set are either terminal additions (incremental scenario), terminal removals (decremental scenario), or both (fully dynamic scenario). Our task here is twofold. We want to support updates in sublinear @math time, and keep the approximation factor of the algorithm as small as possible. We show that we can maintain a @math -approximate Steiner tree of a general graph in @math time per terminal addition or removal. Here, @math denotes the stretch of the metric induced by @math . For planar graphs we achieve the same running time and the approximation ratio of @math . Moreover, we show faster algorithms for incremental and decremental scenarios. Finally, we show that if we allow higher approximation ratio, even more efficient algorithms are possible. In particular we show a polylogarithmic time @math -approximate algorithm for planar graphs. One of the main building blocks of our algorithms are dynamic distance oracles for vertex-labeled graphs, which are of independent interest. We also improve and use the online algorithms for the Steiner tree problem.