Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also 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

Fast and Deterministic Approximations for k-Cut

In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut

Dial a Ride from k-forest

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying at most $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We prove that an $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & Raghavachari; our results give a different proof of a similar approximation guarantee--in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200

Online Steiner Tree with Deletions

In the online Steiner tree problem, the input is a set of vertices that appear one-by-one, and we have to maintain a Steiner tree on the current set of vertices. The cost of the tree is the total length of edges in the tree, and we want this cost to be close to the cost of the optimal Steiner tree at all points in time. If we are allowed to only add edges, a tight bound of $\Theta(\log n)$ on the competitiveness is known. Recently it was shown that if we can add one new edge and make one edge swap upon every vertex arrival, we can maintain a constant-competitive tree online. But what if the set of vertices sees both additions and deletions? Again, we would like to obtain a low-cost Steiner tree with as few edge changes as possible. The original paper of Imase and Waxman had also considered this model, and it gave a greedy algorithm that maintained a constant-competitive tree online, and made at most $O(n^{3/2})$ edge changes for the first $n$ requests. In this paper give the following two results. Our first result is an online algorithm that maintains a Steiner tree only under deletions: we start off with a set of vertices, and at each time one of the vertices is removed from this set: our Steiner tree no longer has to span this vertex. We give an algorithm that changes only a constant number of edges upon each request, and maintains a constant-competitive tree at all times. Our algorithm uses the primal-dual framework and a global charging argument to carefully make these constant number of changes. We then study the natural greedy algorithm proposed by Imase and Waxman that maintains a constant-competitive Steiner tree in the fully-dynamic model (where each request either adds or deletes a vertex). Our second result shows that this algorithm makes only a constant number of changes per request in an amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc

Thresholded Covering Algorithms for Robust and Max-Min Optimization

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? Feige et al. and Khandekar et al. 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?".Comment: 24 page

Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets

Consider the following problem: given a set system (U,I) and an edge-weighted graph G = (U, E) on the same universe U, find the set A in I such that the Steiner tree cost with terminals A is as large as possible: "which set in I is the most difficult to connect up?" This is an example of a max-min problem: find the set A in I such that the value of some minimization (covering) problem is as large as possible. In this paper, we show that for certain covering problems which admit good deterministic online algorithms, we can give good algorithms for max-min optimization when the set system I is given by a p-system or q-knapsacks or both. This result is similar to results for constrained maximization of submodular functions. Although many natural covering problems are not even approximately submodular, we show that one can use properties of the online algorithm as a surrogate for submodularity. Moreover, we give stronger connections between max-min optimization and two-stage robust optimization, and hence give improved algorithms for robust versions of various covering problems, for cases where the uncertainty sets are given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and http://arxiv.org/abs/0912.1045 appeared in ICALP 201