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27
Solving connectivity problems parameterized by treewidth in single exponential time (Extended Abstract)
, 2011
"... For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input g ..."
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Cited by 34 (7 self)
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For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best–known algorithms were naive dynamic programming schemes running in at least tw tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c tw V  O(1) time Monte Carlo algorithms for most connectivitytype problems, including HAMILTONIAN PATH, STEINER TREE, FEEDBACK VERTEX SET and CONNECTED DOMINATING SET. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and Hminorfree graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like CYCLE PACKING.
Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
, 2009
"... We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orien ..."
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Cited by 18 (4 self)
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We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 12 (6 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
Contraction decomposition in Hminorfree graphs and algorithmic applications
 the 43rd ACM Symposium on Theory of Computing (STOC’11
, 2011
"... We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, genera ..."
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Cited by 9 (3 self)
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We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, generalizing previous such decompositions for planar and boundedgenus graphs, and solving the main open problem in this area (posed at SODA 2007). Our decomposition can be computed in polynomial time, resulting in a general framework for approximation algorithms, particularly PTASs (with k ≈ 1/ε), and fixedparameter algorithms, for problems closed under contractions in graphs excluding a fixed minor. For example, our approximation framework gives the first PTAS for TSP in weighted Hminorfree graphs, solving a decadeold open problem of Grohe; and gives another fixedparameter algorithm for kcut in
Prizecollecting Steiner Problems on Planar Graphs
"... In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF), and more generally Submodular PrizeCollecting Steiner Forest (SPCSF), on planar graphs (and also on boundedgenus graphs) to the ..."
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Cited by 9 (2 self)
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In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF), and more generally Submodular PrizeCollecting Steiner Forest (SPCSF), on planar graphs (and also on boundedgenus graphs) to the corresponding problem on graphs of bounded treewidth. More precisely, for each of the mentioned problems, an αapproximation algorithm for the problem on graphs of bounded treewidth implies an (α + ɛ)approximation algorithm for the problem on planar graphs (and also boundedgenus graphs), for any constant ɛ> 0. PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth and hence we obtain a PTAS for these problems on planar graphs and boundedgenus graphs. In
An efficient polynomialtime approximation scheme for Steiner forest in planar graphs
, 2012
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Improved Steiner tree algorithms for bounded treewidth
 IN: IWOCA’11: REVISED SELECTED PAPERS OF THE 22ND INTERNATIONAL WORKSHOP ON COMBINATORIAL ALGORITHMS, LECTURE NOTES IN COMPUTER SCIENCE
, 2011
"... We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in O(B 2 tw+2 · tw · V ) time, where tw is the graph’s treewidth and the Bell number Bk is the number of partitions of a kelement set. This is a linear time algorithm for graphs with fixed ..."
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Cited by 7 (1 self)
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We propose a new algorithm that solves the Steiner tree problem on graphs with vertex set V to optimality in O(B 2 tw+2 · tw · V ) time, where tw is the graph’s treewidth and the Bell number Bk is the number of partitions of a kelement set. This is a linear time algorithm for graphs with fixed treewidth and a polynomial algorithm for tw = O(log V  / log log V ). While being faster than the previously known algorithms, our thereby used coloring scheme can be extended to give new, improved algorithms for the prizecollecting Steiner tree as well as the kcardinality tree problems.
A Polynomialtime Approximation Scheme for Planar Multiway Cut
"... Given an undirected graph with edge lengths and a subset of nodes (called the terminals), a multiway cut (also called a multiterminal cut) problem asks for a subset of edges with minimum total length and whose removal disconnects each terminal from the others. It generalizes the min stcut problem ..."
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Cited by 6 (2 self)
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Given an undirected graph with edge lengths and a subset of nodes (called the terminals), a multiway cut (also called a multiterminal cut) problem asks for a subset of edges with minimum total length and whose removal disconnects each terminal from the others. It generalizes the min stcut problem but is NPhard for planar graphs and APXhard for general graphs. We give a polynomialtime approximation scheme for this problem on planar graphs. We prove the result by building a novel “spanner ” for multiway cut on planar graphs which is of independent interest.
PARAMETERIZED COMPLEXITY OF ARCWEIGHTED DIRECTED STEINER PROBLEMS
, 2011
"... We start a systematic parameterized computational complexity study of three NPhard network design problems on arcweighted directed graphs: directed Steiner tree, strongly connected Steiner subgraph, and directed Steiner network. We investigate their parameterized complexities with respect to the t ..."
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Cited by 6 (0 self)
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We start a systematic parameterized computational complexity study of three NPhard network design problems on arcweighted directed graphs: directed Steiner tree, strongly connected Steiner subgraph, and directed Steiner network. We investigate their parameterized complexities with respect to the three parameterizations: “number of terminals," “an upper bound on the size of the connecting network,” and the combination of these two. We achieve several parameterized hardness results as well as some fixedparameter tractability results, in this way extending previous results of Feldman and Ruhl [SIAM J. Comput., 36 (2006), pp. 543–561].
Prizecollecting Network Design on Planar Graphs
, 2010
"... In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF) and more generally Submodular PrizeCollecting Steiner Forest (SPCSF) on planar graphs (and more generally boundedgenus graphs) ..."
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Cited by 5 (3 self)
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In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting Steiner Tree (PCST), PrizeCollecting Steiner Forest (PCSF) and more generally Submodular PrizeCollecting Steiner Forest (SPCSF) on planar graphs (and more generally boundedgenus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any αapproximation algorithm for these problems on graphs of bounded treewidth gives an (α + ɛ)approximation algorithm for these problems on planar graphs (and more generally boundedgenus graphs), for any constant ɛ> 0. Since PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth using dynamic programming, we obtain PTASs for these problems on planar graphs and boundedgenus graphs. In contrast, we show PCSF is APXhard to approximate on seriesparallel graphs, which are planar graphs of treewidth at most 2. This result is interesting on its own because it gives the first provable hardness separation between prizecollecting and nonprizecollecting (regular) versions of the problems: regular Steiner Forest is known to be polynomially solvable on seriesparallel graphs and admits a PTAS on graphs of bounded treewidth. An analogous hardness result can be shown for Euclidian PCSF. This ends the common belief that prizecollecting variants should not add any new hardness to the problems.