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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.
Parameterized singleexponential time polynomial space algorithm for Steiner Tree
"... In the Steiner tree problem, we are given as input a connected nvertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimumweight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97k ·n4 · ..."
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In the Steiner tree problem, we are given as input a connected nvertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimumweight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97k ·n4 · log W) using O(n3 · log nW · log k) space. This is the first singleexponential time, polynomialspace FPT algorithm for the weighted Steiner Tree problem.
DIRECTED GRAPHS: FIXEDPARAMETER TRACTABILITY & BEYOND
, 2014
"... Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The pa ..."
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Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NPhard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixedparameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) · nO(1) time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) ·nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively