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46
Approximation algorithms via contraction decomposition
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge ..."
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Cited by 36 (7 self)
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We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contractionclosed problems (whose optimal solution only improves under contraction), a much more general class than minorclosed problems. We prove that any contractionclosed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight cedgeconnected submultigraph on boundedgenus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general Hminorfree graphs.
Computing shortest nontrivial cycles on orientable surfaces of bounded genus in almost linear time
 In SOCG ’06: Proc. 22nd Symp. Comput. Geom
, 2006
"... We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon ..."
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Cited by 35 (0 self)
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We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon the currentbest O(n 3/2)time algorithm by Cabello and Mohar (ESA 2005). Our algorithm uses universalcover constructions to find short cycles and makes extensive use of existing tools from the field. 1
Multiple source shortest paths in a genus g graph
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms
"... We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source ..."
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Cited by 35 (12 self)
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We give an O(g2n log n) algorithm to represent the shortest path tree from all the vertices on a single specified face f in a genus g graph. From this representation, any query distance from a vertex in f can be obtained in O(log n) time. The algorithm uses a kinetic data structure, where the source of the tree iteratively movesacrossedgesinf. In addition, we give applications using these shortest path trees in order to compute the shortest noncontractible cycle and the shortest nonseparating cycle embedded on an orientable 2manifold in O(g3n log n) time. 1
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 33 (11 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Distributed coverage verification in sensor networks without location information
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2008
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Tightening NonSimple Paths and Cycles on Surfaces
 SUBMITTED TO SIAM JOURNAL ON COMPUTING
"... We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity ..."
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Cited by 27 (10 self)
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We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity n, genus g ≥ 2, and no boundary, we construct in O(gn log n) time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity k in O(gnk) time, or the shortest cycle homotopic to a given cycle of complexity k in O(gnk log(nk)) time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.
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 27 (7 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.
Optimal Pants Decompositions and Shortest Homotopic Cycles On An Orientable Surface
, 2007
"... We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solv ..."
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Cited by 24 (2 self)
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We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertexedge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal: each cycle is as short as possible in its homotopy class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this paper are polynomial, assuming uniformity of the weights of the vertexedge graph of the surface.
Graph and map isomorphism and all polyhedral embeddings in linear time
 IN PROCEEDINGS OF THE 40TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2008
"... For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. Th ..."
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Cited by 20 (6 self)
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For every surface S (orientable or nonorientable), we give a linear time algorithm to test the graph isomorphism of two graphs, one of which admits an embedding of facewidth at least 3 into S. This improves a previously known algorithm whose time complexity is n O(g), where g is the genus of S. This is the first algorithm for which the degree of polynomial in the time complexity does not depend on g. The above result is based on two linear time algorithms, each of which solves a problem that is of independent interest. The first of these problems is the following one. Let S be a fixed surface. Given a graph G and an integer k≥3, we want to find an embedding of G in S of face width at least k, or conclude that such an embedding does not exist. It is known that this problem is NPhard when the surface is not fixed. Moreover, if there is an embedding, the algorithm can give all embeddings of facewidth at least k, up to Whitney equivalence. Here, the facewidth of an embedded graph G is the minimum number of points of G in which some noncontractible closed curve in the surface intersects the graph. In the proof of the above algorithm, we give a simpler proof and a better bound for the theorem by Mohar and Robertson concerning the number of polyhedral embeddings of 3connected graphs.