An instance of the k-Steiner forest problem consists of an undirected graph G = (V, E), the edges of which are associated with non-negative costs, and a collection D = {(s1, t1),..., (sd, td)} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest F ⊆ G connects a demand (si, ti) when it contains an si-ti path. Given a requirement parameter k ≤ |D|, the goal is to find a minimum cost forest that connects at least k demands in D. This problem has recently been studied by Hajiaghayi and Jain [SODA ’06], whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k-subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n 2/3, √ d} · log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

Abstract. In this paper we design an iterative rounding approach for the classic prize-collecting Steiner forest problem and more generally the prize-collecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable in a basic feasible solution which is at least one-third-integral resulting a 3-approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prize-collecting Steiner forest and thus prize-collecting survivable Steiner network design. This especially answers negatively the previous belief that one might be able to obtain an approximation factor better than 3 for these problems using a natural iterative rounding approach. Our structural result is extending the celebrated iterative rounding approach of Jain [13] by using several new ideas some from more complicated linear algebra. The approach of this paper can be also applied to get a constant factor (bicriteria-)approximation algorithm for degree constrained prize-collecting network design problems. We emphasize that though in theory we can prove existence of only an LP variable of at least one-third-integral, in practice very often in each iteration there exists a variable of integral or almost integral which results in a much better approximation factor than provable factor 3 in this paper (see patent application [11]). This is indeed the advantage of our algorithm in this paper over previous approximation algorithms for prize-collecting Steiner forest with the same or slightly better provable approximation factors. 1

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF) and more generally Submodular Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally bounded-genus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any α-approximation algorithm for these problems on graphs of bounded treewidth gives an (α + ɛ)-approximation algorithm for these problems on planar graphs (and more generally bounded-genus graphs), for any constant ɛ> 0. Since PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth using dynamic programming, we obtain PTASs for these problems on planar graphs and bounded-genus graphs. In contrast, we show PCSF is APX-hard to approximate on series-parallel graphs, which are planar graphs of treewidth at most 2. This result is interesting on its own because it gives the first provable hardness separation between prize-collecting and non-prize-collecting (regular) versions of the problems: regular Steiner Forest is known to be polynomially solvable on series-parallel graphs and admits a PTAS on graphs of bounded treewidth. An analogous hardness result can be shown for Euclidian PCSF. This ends the common belief that prize-collecting variants should not add any new hardness to the problems.

Abstract. We consider an optimization problem consisting of an undirected graph, with cost and profit functions defined on all vertices. The goal is to find a connected subset of vertices with maximum total profit, whose total cost does not exceed a given budget. The best result known prior to this work guaranteed a (2, O(log n)) bicriteria approximation, i.e. the solution’s profit is at least a frac-1 tion of of an optimum solution respecting the budget, while its cost O(log n) is at most twice the given budget. We improve these results and present a bicriteria tradeoff that, given any ε ∈ (0, 1], guarantees a (1 + ε, O ( 1 log n))-ε approximation. 1

In the k-2VC problem, we are given an undirected graph G with edge costs and an integer k; the goal is to find a minimum-cost 2-vertex-connected subgraph of G containing at least k vertices. A slightly more general version is obtained if the input also specifies a subset S ⊆ V of terminals and the goal is to find a subgraph containing at least k terminals. Closely related to the k-2VC problem, and in fact a special case of it, is the k-2EC problem, in which the goal is to find a minimum-cost 2-edge-connected subgraph containing k vertices. The k-2EC problem was introduced by Lau et al. [22], who also gave a poly-logarithmic approximation for it. No previous approximation algorithm was known for the more general k-2VC problem. We describe an O(log n · log k) approximation for the k-2VC problem.

Given an edge-weighted undirected graph G with a specified set of terminals, let the density of any subgraph be the ratio of its weight/cost to the number of terminals it contains. If G is 2-connected, does it contain smaller 2-connected subgraphs of density comparable to that of G? We answer this question in the affirmative by giving an algorithm to prune G and find such subgraphs of any desired size, at the cost of only alogarithmic increase in density (plus a small additive factor). We apply the pruning techniques to give algorithms for two NP-Hard problems on finding large 2-vertex-connectedsubgraphsoflowcost;nopreviousapproximationalgorithmwasknownforeither problem. Inthe k-2VC problem,wearegivenanundirectedgraph G withedgecostsand an integer k; thegoalistofindaminimum-cost2-vertex-connected subgraph of G containing at least k vertices. In the Budget-2VC problem, we are given the graph G with edge costs, and a budget B; the goal is to find a 2-vertex-connected subgraph H of G with total edge cost at most B that maximizes the number of vertices in H. We describe an O(lognlogk) approximation for the k-2VC problem, and a bicriteria approximation for the Budget-2VC problem that gives an O ( 1 ǫ log2 n) approximation, while violating the budget by a factor of atmost 3 + ǫ.

An instance of the k-generalized connectivity problem consists of an undirected graph G = (V, E), whose edges are associated with non-negative costs, and a collection D = {(S1, T1),..., (Sd, Td)} of distinct demands, each of which comprises a pair of disjoint vertex sets. We say that a subgraph H ⊆ G connects a demand (Si, Ti) when it contains a path with one endpoint in Si and the other in Ti. Given an integer parameter k, the goal is to identify a minimum cost subgraph that connects at least k demands in D. Alon, Awerbuch, Azar, Buchbinder and Naor (SODA ’04) seem to have been the first to consider the generalized connectivity paradigm as a unified machinery for incorporating multiplechoice decisions into network formation settings. Their main contribution in this context was to devise a multiplicative-update online algorithm for computing log-competitive fractional solutions, and to propose provably-good rounding procedures for important special cases. Nevertheless, approximating the generalized connectivity problem in its unconfined form, where one makes no structural assumptions about the underlying graph and collection of demands, has remained an open question up until a recent O(log 2 n log 2 d) approximation due to Chekuri, Even, Gupta and Segev (SODA ’08). Unfortunately, the latter result does not extend to connecting a pre-specified number of demands. Furthermore, even the simpler case of singleton demands has been established as a challenging computational task, when Hajiaghayi and Jain (SODA ’06) related its inapproximability to that of dense k-subgraph. In this paper, we present the first non-trivial approximation algorithm for k-generalized connectivity, which is derived by synthesizing several techniques originating in probabilistic embeddings of finite metrics, network design, and randomization. Specifically, our algorithm constructs, with constant probability, a feasible subgraph whose cost is within a factor of O(n 2/3 · polylog(n, k)) of optimal. We believe that the fundamental approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.

I give an account of the paper [HJ06] on the prize-collecting generalized Steiner tree problem.

We study approximation algorithms for generalized network design where the cost of an edge depends on the identities of the demands using it (as a monotone subadditive function). Our main result is that even a very special case of this problem cannot be approximated to within a factor 2 log1−ε |D| if D is the set of demands

In a Content Distribution Network application, we have a set of servers and a set of clients to be connected to the servers. Often there are a few server types and a hard budget constraint on the number of deployed servers of each type. The simplest goal here is to deploy a set of servers subject to these budget constraints in order to minimize the sum of client connection costs. These connection costs often satisfy metricity, since they are typically proportional to the distance between a client and a server within a single autonomous system. A special case of the problem where there is only one server type is the well-studied k-median problem. In this paper, we consider the problem with two server types, called the budgeted red-blue median problem, which is interesting in its own right. We show, somewhat surprisingly, that running a single-swap local search for each server type simultaneously, yields a constant factor approximation for this case. Its analysis is however quite non-trivial compared to that of the k-median problem (Arya et al., 2004; Gupta and Tangwongsan, 2008). Later we show that the same algorithm yields a constant approximation for the prize-collecting version of the budgeted red-blue median problem where each client can potentially be served with an alternative cost via a different vendor. In the process, we also improve the approximation factor for the prize-collecting k-median problem from 4 (Charikar et al., 2001) to 3 + ɛ, which matches the current best approximation factor for the k-median problem.