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27
Euclidean Prizecollecting Steiner Forest
, 2009
"... In this paper, we consider Steiner forest and its generalizations, prizecollecting Steiner forest and kSteiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomialtime approxi ..."
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In this paper, we consider Steiner forest and its generalizations, prizecollecting Steiner forest and kSteiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomialtime approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a wellmotivated case, i.e., the multiplicative case, of prizecollecting and budgeted Steiner forest. The ideas used in the algorithm may have applications in design of a broad class of bicriteria PTASs. At the end, we demonstrate why PTASs for these problems can be hard in the general Euclidean case (and thus for PTASs we cannot go beyond the multiplicative case).
LinearTime Computation of a Linear Problem Kernel for Dominating Set on Planar Graphs
, 2011
"... Over the last years, reduction to a problem kernel, or kernelization for short, has developed into a very active research area within parameterized complexity analysis and the algorithmics of NPhard problems in general [2]. In a nutshell, a kernelization algorithm transforms in polynomial time an i ..."
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Cited by 4 (2 self)
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Over the last years, reduction to a problem kernel, or kernelization for short, has developed into a very active research area within parameterized complexity analysis and the algorithmics of NPhard problems in general [2]. In a nutshell, a kernelization algorithm transforms in polynomial time an instance of a (typically NPhard) problem to an equivalent instance whose size is bounded by a function of a parameter. Nowadays, it has become a standard challenge to minimize the size of problem kernels. Consider the following examples: 1. For Feedback Vertex Set (given an undirected graph G and a positive integer k, find at most k vertices whose deletion destroys all cycles in G), there first has been an O(k 11)vertex problem kernel [3], improved to an O(k 3)vertex problem kernel and finally to an O(k 2)vertex problem kernel [5] 2. For Dominating Set on planar graphs (given an undirected planar graph G and a positive integer k, find at most k vertices such that each other vertex in G has at least one neighbor in the set of selected vertices), there first was a 335kvertex problem kernel [1], which was further refined into a 67kvertex problem kernel [4], both computable in O(n 3) time. From the viewpoint of practical relevance, however, also the running times of kernelization algorithms have to be optimized. We show an O(k)size problem kernel for Dominating Set in planar graphs that is computable in O(n) time.
SubexponentialTime Parameterized Algorithm for Steiner Tree on Planar Graphs ∗
"... 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 orde ..."
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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.
The Steiner Multigraph Problem: Wildlife Corridor Design for Multiple Species
"... The conservation of wildlife corridors between existing habitat preserves is important for combating the effects of habitat loss and fragmentation facing species of concern. We introduce the Steiner Multigraph Problem to model the problem of minimumcost wildlife corridor design for multiple species ..."
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The conservation of wildlife corridors between existing habitat preserves is important for combating the effects of habitat loss and fragmentation facing species of concern. We introduce the Steiner Multigraph Problem to model the problem of minimumcost wildlife corridor design for multiple species with different landscape requirements. This problem can also model other analogous settings in wireless and social networks. As a generalization of Steiner forest, the goal is to find a minimumcost subgraph that connects multiple sets of terminals. In contrast to Steiner forest, each set of terminals can only be connected via a subset of the nodes. Generalizing Steiner forest in this way makes the problem NPhard even when restricted to two pairs of terminals. However, we show that if the node subsets have a nested structure, the problem admits a fixedparameter tractable algorithm in the number of terminals. We successfully test exact and heuristic solution approaches on a wildlife corridor instance for wolverines and lynx in western Montana, showing that though the problem is computationally hard, heuristics perform well, and provably optimal solutions can still be obtained. 1
Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions
 In SODA
, 2014
"... Given a vertexweighted 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 NPhard, but ..."
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Given a vertexweighted 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 NPhard, 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) algorithm for planar SCSS, which is an improvement of a factor of O( k) in the exponent over the algorithm 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 computable function f, unless the Exponential Time Hypothesis (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 treewidthbased 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 gridlike fashion to tightly control the number of terminals in the created instance.
Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
, 2010
"... This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialt ..."
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This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialtime) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N PHard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing faulttolerant networks. Two vertices u, v in a network are said to be kedgeconnected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are kvertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning
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
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.
Correlation Clustering and Twoedgeconnected Augmentation for Planar Graphs
"... In correlation clustering, the input is a graph with edgeweights, where every edge is labelled either + or − according to similarity of its endpoints. The goal is to produce a partition of the vertices that disagrees with the edge labels as little as possible. In twoedgeconnected augmentation, th ..."
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In correlation clustering, the input is a graph with edgeweights, where every edge is labelled either + or − according to similarity of its endpoints. The goal is to produce a partition of the vertices that disagrees with the edge labels as little as possible. In twoedgeconnected augmentation, the input is a graph with edgeweights and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are twoedgeconnected in R ∪ S. For planar graphs, we prove that correlation clustering reduces to twoedgeconnected augmentation, and that both problems have a polynomialtime approximation scheme.