A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Klein, Philip N.

doi:10.1137/060649562

Cited by 64 publications

(59 citation statements)

References 35 publications

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“…This improvement resolves a 15 year old conjecture of Rao and Smith, and matches for Euclidean spaces the bound known for a broad class of planar graphs [Kle08]. …”

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confidence: 69%

A Linear Time Approximation Scheme for Euclidean TSP

Bartal

Gottlieb

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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“…This improvement resolves a 15 year old conjecture of Rao and Smith, and matches for Euclidean spaces the bound known for a broad class of planar graphs [Kle08]. …”

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confidence: 69%

A Linear Time Approximation Scheme for Euclidean TSP

Bartal

Gottlieb

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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“…In addition to our novel bootstrapping approach, we also need to first construct a spanner for planar group Steiner tree. In particular, deciding which terminal in a group is the one to participate in an optimal solution makes this task much harder than previous and recent approaches to construct spanners and thus obtain PTASs for planar TSP by Klein [26], subset TSP by Klein [25], Steiner tree by Borradaile, Klein, and Mathieu [8], and Steiner forest by Bateni, Hajiaghayi, and Marx [5] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [16]). Last but not least, we show planar group Steiner forest, a slight generalization of planar group Steiner tree in which the goal is to find a forest of minimum length that connects pairs of given group terminals is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge.…”

Section: Introductionmentioning

confidence: 99%

A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting

Bateni

Demaine

Hajiaghayi

et al. 2016

Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing

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We present the first polynomial-time approximation scheme (PTAS), i.e., (1 + ε)-approximation algorithm for any constant ε > 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(log n(log log n) O(1) ). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing lightweight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems.Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which * Supported in part by NSF Grant CCF-1161626, NSF grant IIS-1546108, and DARPA/AFOSR GRAPHS grant FA9550-12-1-0423. reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution.Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2 + ε)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge.

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“…Not being able to solve Steiner forest optimally on boundedtreewidth graphs is not an unavoidable obstacle for obtaining a PTAS on planar graphs: the technique of Klein [21], and Demaine, Hajiaghayi and Mohar [13] can still be applied when we have a PTAS for graphs of bounded treewidth. In Section 5, we demonstrate such a PTAS: THEOREM 5.…”

Section: Dmentioning

confidence: 99%

Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth

Bateni

Hajiaghayi

Marx

2010

Proceedings of the Forty-Second ACM Symposium on Theory of Computing

114

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We give the first polynomial-time 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 prize-collecting 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 NP-hard 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 series-parallel 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. * For the omitted proofs and further discussion, refer to the full version of the paper [6].

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A Linear-Time Approximation Scheme for TSP in Undirected Planar Graphs with Edge-Weights

Abstract: We give an algorithm requiring O(c 1/ǫ 2 n) time to find an ǫ-optimal traveling salesman tour in the shortest-path metric defined by an undirected planar graph with nonnegative edgelengths. For the case of all lengths equal to 1, the time required is O(c 1/ǫ n).

Cited by 64 publications

References 35 publications

A Linear Time Approximation Scheme for Euclidean TSP

A Linear Time Approximation Scheme for Euclidean TSP

A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting

Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth

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