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Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth
, 2010
"... We give the first polynomialtime approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called priz ..."
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Cited by 28 (8 self)
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We give the first polynomialtime approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prizecollecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NPhard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for seriesparallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs.
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.
The Maximum Weight Connected Subgraph Problem
, 2013
"... The Maximum (Node) Weight Connected Subgraph Problem (MWCS) searches for a connected subgraph with maximum total weight in a nodeweighted (di)graph. In this work we introduce a new integer linear programming formulation built on node variables only, which uses new constraints based on nodeseparat ..."
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Cited by 3 (1 self)
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The Maximum (Node) Weight Connected Subgraph Problem (MWCS) searches for a connected subgraph with maximum total weight in a nodeweighted (di)graph. In this work we introduce a new integer linear programming formulation built on node variables only, which uses new constraints based on nodeseparators. We theoretically compare its strength to previously used MIP models in the literature and study the connected subgraph polytope associated with our new formulation. In our computational study we compare branchandcut implementations of the new model with two models recently proposed in the literature: one of them using the transformation into the PrizeCollecting Steiner Tree problem, and the other one working on the space of node variables only. The obtained results indicate that the new formulation outperforms the previous ones in terms of the running time and in terms of the stability with respect to variations of node weights.
Fast Algorithms for Structured Sparsity (ICALP 2015 Invited Tutorial)
"... Sparsity has become an important tool in many mathematical sciences such as statistics, machine learning, and signal processing. While sparsity is a good model for data in many applications, data often has additional structure that goes beyond the notion of “standard ” sparsity. In many cases, we ca ..."
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Sparsity has become an important tool in many mathematical sciences such as statistics, machine learning, and signal processing. While sparsity is a good model for data in many applications, data often has additional structure that goes beyond the notion of “standard ” sparsity. In many cases, we can represent this additional information in a structured sparsity model. Recent research has shown that structured sparsity can improve the sample complexity in several applications such as compressive sensing and sparse linear regression. However, these improvements come at a computational cost, as the data needs to be “fitted ” so it satisfies the constraints specified by the sparsity model. In this survey, we introduce the concept of structured sparsity, explain the relevant algorithmic challenges, and briefly describe the best known algorithms for two sparsity models. On the way, we demonstrate that structured sparsity models are inherently combinatorial structures, and employing structured sparsity often leads to interesting algorithmic problems with strong connections to combinatorial optimization and discrete algorithms. We also state several algorithmic open problems related to structured sparsity. 1
A Polynomialtime Bicriteria Approximation Scheme for Planar Bisection
, 2015
"... Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar ..."
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Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let W be the total weight of all nodes in a planar graph G. For any constant ε> 0, our algorithm outputs a bipartition of the nodes such that each part weighs at most W/2+ε and the total cost of edges crossing the partition is at most (1+ε) times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was O(log n). Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.
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
Claire Mathieu
"... We give a randomizedO(npolylogn)time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed > 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1+ )approximation to the minimumlength forest th ..."
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We give a randomizedO(npolylogn)time approximation scheme for the Steiner forest problem in the Euclidean plane. For every fixed > 0 and given n terminals in the plane with connection requests between some pairs of terminals, our scheme finds a (1+ )approximation to the minimumlength forest that connects every requested pair of terminals. 1