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SubexponentialTime Parameterized Algorithm for Steiner Tree on Planar Graphs
, 2013
"... The wellknown bidimensionality theory provides a method for designing fast, subexponentialtime parameterized algorithms for a vast number of NPhard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in ord ..."
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The wellknown bidimensionality theory provides a method for designing fast, subexponentialtime parameterized algorithms for a vast number of NPhard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in order to apply the bidimensionality framework the considered problem needs to fulfill a special density property. Some wellknown problems do not have this property, unfortunately, with probably the most prominent and important example being the Steiner Tree problem. Hence the question whether a subexponentialtime parameterized algorithm for Steiner Tree on planar graphs exists has remained open. In this paper, we answer this question positively and develop an algorithm running in O(2 O((k log k)2/3) n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph. Our algorithm does not rely on tools from bidimensionality theory or graph minors theory, apart from Baker’s classical approach. Instead, we introduce new tools and concepts to the study of the parameterized complexity of problems on sparse graphs.
Leeuwen, Network sparsification for steiner problems on planar and boundedgenus graphs
 Proc. 55th FOCS, abs/1306.6593
, 2013
"... We propose polynomialtime algorithms that sparsify planar and boundedgenus graphs while preserving optimal or nearoptimal solutions to Steiner problems. Our main contribution is a polynomialtime algorithm that, given an unweighted graph G embedded on a surface of genus g and a designated face f ..."
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We propose polynomialtime algorithms that sparsify planar and boundedgenus graphs while preserving optimal or nearoptimal solutions to Steiner problems. Our main contribution is a polynomialtime algorithm that, given an unweighted graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k, uncovers a set F ⊆ E(G) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f. We apply this general theorem to prove that: • given an unweighted graph G embedded on a surface of genus g and a terminal set S ⊆ V (G), one can in polynomial time find a set F ⊆ E(G) that contains an optimal Steiner tree T for S and that has size polynomial in g and E(T); • an analogous result holds for an optimal Steiner forest for a set S of terminal pairs; • given an unweighted planar graph G and a terminal set S ⊆ V (G), one can in polynomial time find a set F ⊆ E(G) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different termi
Dynamic Programming for Hminorfree Graphs
"... We provide a framework for the design and analysis of dynamic programming algorithms for Hminorfree graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2O(k·log k) · nO(1) steps, with n being the number ..."
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We provide a framework for the design and analysis of dynamic programming algorithms for Hminorfree graphs with branchwidth at most k. Our technique applies to a wide family of problems where standard (deterministic) dynamic programming runs in 2O(k·log k) · nO(1) steps, with n being the number of vertices of the input graph. Extending the approach developed by the same authors for graphs embedded in surfaces, we introduce a new type of branch decomposition for Hminorfree graphs, called an Hminorfree cut decomposition, and we show that they can be constructed in Oh(n 3) steps, where the hidden constant depends exclusively on H. We show that the separators of such decompositions have connected packings whose behavior can be described in terms of a combinatorial object called `triangulation. Our main result is that when applied on Hminorfree cut decompositions, dynamic programming runs in 2Oh(k) ·nO(1) steps. This broadens substantially the class of problems that can be solved deterministically in singleexponential time for Hminorfree graphs.
Subexponential Parameterized Odd Cycle Transversal on Planar Graphs
"... In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with ..."
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In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, and the objective is to determine whether there exists a vertex set O in G of size at most k such that G \ O is bipartite. Reed, Smith, and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3 k n O(1). Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time cannot be improved to 2 o(k) n O(1). We show that OCT admits a randomized algorithm running in O(n O(1) + 2 O( √ k log k) n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(2 O(k log k) n) time. This improves over an algorithm of Fiorini et