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Approximating Rooted Steiner Networks
"... The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APXhard. The problem with one root and many sinks is as hard to approximate as the dire ..."
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The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APXhard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(k ɛ) hardness bound for the rooted kconnectivity problem in undirected graphs; this addresses a recent open question of Khanna. As a consequence, we also obtain the Ω(k ɛ) hardness of the undirected subset kconnectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted kconnectivity problem. 1
FixedParameter and Approximation Algorithms: A New Look
"... Abstract. A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k e ..."
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Abstract. A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For wellknown intractable problems such as the W[1]hard Clique and W[2]hard Set Cover problems, the natural question is whether we can get any FPTapproximation. It is widely believed that both Clique and SetCover admit no FPT ρapproximation algorithm, for any increasing function ρ. However, to the best of our knowledge, there has been no progress towards proving this conjecture. Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) [18] and the Projection Games Conjecture (PGC) [27], we make the first progress towards proving this conjecture by showing that – Under the ETH and PGC, there exist constants F1, F2> 0 such that the Set Cover problem does not admit a FPT approximation algorithm with ratio k F1 k in 2 F2 · poly(N, M) time, where N is the size of the universe and M is the number of sets. – Unless NP ⊆ SUBEXP, for every 1> δ> 0 there exists a constant F (δ)> 0 such that Clique has no FPT cost approximation with ratio k 1−δ in 2 kF · poly(n) time, where n is the number of vertices in the graph. In the second part of the paper we consider various W[1]hard problems
Parameterized complexity and kernel bounds for hard planning problems
 Algorithms and Complexity, 8th International Conference, CIAC 2013
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Parameterized complexity of directed steiner tree on sparse graphs
, 2012
"... Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the para ..."
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Abstract. We study the parameterized complexity of the directed variant of the classical Steiner Tree problem on various classes of directed sparse graphs. While the parameterized complexity of Steiner Tree parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of nonterminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]hard on general graphs, and hence unlikely to be fixed parameter tractable (FPT). The undirected Steiner Tree problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for Directed Steiner Tree (DST) on sparse graphs parameterized by the number of nonterminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for Dominating Set on sparse undirected graphs. These algorithms are asymptotically optimal. 1
Directed Nowhere Dense Classes of Graphs
, 2012
"... Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of g ..."
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Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of graphs on which these problems become tractable. While in the case of undirected graphs, there is a rich structure theory which can be used to develop tractable algorithms for these problems on large classes of undirected graphs, such a theory is much less developed for directed graphs. Many attempts to identify structure properties of directed graphs tailored towards algorithmic applications have focussed on a directed analogue of undirected treewidth. These attempts have proved to be successful in the development
Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions
 In SODA
, 2014
"... Given a vertexweighted directed graph G = (V,E) and a set T = {t1, t2,... tk} of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆V of minimum weight such that G[H] contains a ti → t j path for each i 6 = j. The problem is NPhard, but ..."
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Given a vertexweighted directed graph G = (V,E) and a set T = {t1, t2,... tk} of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆V of minimum weight such that G[H] contains a ti → t j path for each i 6 = j. The problem is NPhard, but Feldman and Ruhl (FOCS ’99; SICOMP ’06) gave a novel nO(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs. • Our main algorithmic result is a 2O(k logk) ·nO( k) algorithm for planar SCSS, which is an improvement of a factor of O( k) in the exponent over the algorithm of Feldman and Ruhl. • Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f (k) · no( k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails. The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidthbased techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a gridlike fashion to tightly control the number of terminals in the created instance.