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Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions)
- IN SODA
, 2014
"... Given a vertex-weighted 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 NP-hard, but ..."
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Cited by 2 (1 self)
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Given a vertex-weighted 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 NP-hard, 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) al-gorithm for planar SCSS, which is an improvement of a factor of O( k) in the exponent over the algo-rithm 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 com-putable function f, unless the Exponential Time Hy-pothesis (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 treewidth-based 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 grid-like fashion to tightly control the number of terminals in the created instance.
Parameterized single-exponential time polynomial space algorithm for Steiner Tree
"... In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight 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 n-vertex graph with edge weights in {1, 2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight 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 single-exponential time, polynomial-space FPT algorithm for the weighted Steiner Tree problem.
DIRECTED GRAPHS: FIXED-PARAMETER TRACTABILITY & BEYOND
, 2014
"... Most interesting optimization problems on graphs are NP-hard, implying that (un-less P = NP) there is no polynomial time algorithm that solves all the instances of an NP-hard 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 NP-hard, implying that (un-less P = NP) there is no polynomial time algorithm that solves all the instances of an NP-hard 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 analy-sis 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 com-plexity is to design efficient algorithms for NP-hard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixed-parameter 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 pa-rameterized 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 connec-tivity problems on directed graphs. This research direction has been hitherto relatively