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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
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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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
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 6 (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.
Abstract In this paper, we reduce PrizeCollecting Steiner TSP (PCTSP), PrizeCollecting Stroll (PCS), PrizeCollecting
"... est (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 + )approximation algorithm fo ..."
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est (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 + )approximation algorithm for the problem on planar graphs (and also boundedgenus graphs), for any constant > 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 contrast, we show that PCSF is APXhard to approximate on seriesparallel graphs, which are planar graphs of treewidth at most 2. Apart from ruling out a PTAS for PCSF on planar graphs and bounded treewidth graphs, this result is also interesting since it gives the first provable hardness separa
APPROXIMATION ALGORITHMS FOR SUBMODULAR OPTIMIZATION AND GRAPH PROBLEMS
, 2013
"... In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intracta ..."
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In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intractability of these problems, we focus on algorithms that construct approximate solutions: An approximation algorithm is a polynomialtime algorithm that, for any instance of the problem, it outputs a solution whose value is within a multiplicative factor ρ of the value of the optimal solution for the instance. The quantity ρ is the approximation ratio of the algorithm and we aim to achieve the smallest ratio possible. Our focus in this thesis is on designing approximation algorithms for several combinatorial optimization problems. In the first part of this thesis, we study a class of constrained submodular minimization problems. We introduce a model that captures allocation problems with submodular costs and we give a generic approach for designing approximation algorithms for problems in this model. Our model captures several problems of interest, such as nonmetric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization
A PrimalDual Clustering Technique with Applications in Network Design
, 2011
"... Network design problems deal with settings where the goal is to design a network (i.e., find a subgraph of a given graph) that satisfies certain connectivity requirements. Each requirement is in the form of connecting (or, more generally, providing large connectivity between) a pair of vertices of t ..."
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Network design problems deal with settings where the goal is to design a network (i.e., find a subgraph of a given graph) that satisfies certain connectivity requirements. Each requirement is in the form of connecting (or, more generally, providing large connectivity between) a pair of vertices of the graph. The goal is to find a network of minimum length, and in some cases requirements can be compromised after paying their “penalties. ” These are usually called prizecollecting Steiner network problems. In practical scenarios of physical networking, with cable or fiber embedded in the ground, crossings are rare or nonexistent. Hence, planar instances of network design problems are a natural subclass of interest. We can usually take advantage of this structure to find better performance guarantees. In this thesis, we develop a primaldual clustering technique called “prizecollecting clustering, ” which is used to give improved approximation algorithms for several planar and nonplanar network design problems. The technique is based on a famous moat growing procedure due to Agrawal, Klein, Ravi [AKR95] and Goemans and Williamson [GW95]. It provides a paradigm for clustering the vertices of a graph