We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs. * For the omitted proofs and further discussion, refer to the full version of the paper [6].

Online auction is an essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic secretary problem, is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the multiple-choice secretary problem were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the submodular secretary problem, in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log 2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of subadditive secretary problem, there is noõ( √ n)-competitive algorithm and thus submodular functions are the most general functions to consider for constant competitiveness in our setting. We complement this result by giving a matching O( √ n)-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well. *

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio.In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include:• The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a (1.5 + ε)-approximation streaming algorithm that uses O(n 1.5 ) space for constant ε > 0. • The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a (1.5 + ε)-approximation algorithm with O( √ mn + n) memory per machine for constant ε > 0.By building on our unified approach, we further develop parallel algorithms in the MPC model that give a (1 + ǫ)-approximation to matching and an O(1)-approximation to vertex cover in only O(log log n) MPC rounds and O(n/polylog(n)) memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj et al. [STOC 2018].We obtain our results by a novel combination of two previously disjoint set of techniques, namely randomized composable coresets and edge degree constrained subgraphs (EDCS). We significantly extend the power of these techniques and prove several new structural results. For example, we show that an EDCS is a sparse certificate for large matchings and small vertex covers that is quite robust to sampling and composition. * sassadi@cis.upenn.edu.As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems on large datasets. When dealing with massive data, the entire input graph is orders of magnitude larger than the amount of storage on one processor and hence any algorithm needs to explicitly address this issue. For massive inputs, several different computational models have been introduced, each focusing on certain additional resources needed to solve largescale problems. Some examples include the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReducestyle computation (see Section 2 for a definition of MPC). The target resources in these models are the number of rounds of communication and the local storage on each machine.Given the variety of relevant models, there has been a lot of attention on designing general algorithmic techniques that can be applicable across a wide ran...

Abstract-We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem.We present (2 − )-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier.

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs. * For the omitted proofs and further discussion, refer to the full version of the paper [6].