A note on approximating graph genus

Chen, Jianer; Kanchi, Saroja; Kanevsky, Arkady

doi:10.1016/s0020-0190(97)00203-2

Cited by 17 publications

(17 citation statements)

References 9 publications

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“…] is composed of at most four subpaths of P. 2. ψ i has genus ≤ γ and all marginal edges M i are on ≤ α disjoint (ψ i , P)-intervals.…”

Section: P[s Imentioning

confidence: 99%

“…By Euler's characteristic, there is a trivial O(1)-approximation for sufficiently dense graphs (i. e., of average degree at least 6 + ε, for some fixed ε > 0). For graphs of bounded degree, Chen, Kanchi, and Kanevsky [2] described a simple O( √ n)-approximation, which follows by the fact that graphs of small genus have small balanced vertex-separators. Following the present paper, Chekuri and Sidiropoulos [1] obtained a polynomial-time algorithm which, given a graph G of bounded degree and of genus g, outputs a drawing on a surface of genus O(g 12 log 19/2 n).…”

Section: Introductionmentioning

confidence: 99%

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Makarychev¹,

Nayyeri²,

Sidiropoulos³

2017

Theory of Comput.

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“…] is composed of at most four subpaths of P. 2. ψ i has genus ≤ γ and all marginal edges M i are on ≤ α disjoint (ψ i , P)-intervals.…”

Section: P[s Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Makarychev¹,

Nayyeri²,

Sidiropoulos³

2017

Theory of Comput.

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“…Computing the minimum genus of an abstract graph is NP-hard [111]; moreover, no efficient algorithms are known that approximate the genus within a factor of o( n) [22]. On the other hand, for any constant g, it is possible to compute either an embedding of a given graph on a surface of genus g, or an obstruction to such an embedding, in O(n) time [77,93].…”

Section: Sparse Graphsmentioning

confidence: 99%

Homology Flows, Cohomology Cuts

Chambers

Erickson²,

Nayyeri

2012

SIAM J. Comput.

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We describe the first algorithm to compute maximum flows in surface-embedded graphs in near-linear time. Specifically, given a graph embedded on a surface of genus g, with two specified vertices s and t and integer edge capacities that sum to C, our algorithm computes a maximum (s, t)-flow in O(g 8 n log 2 n log 2 C) time. We also present a combinatorial algorithm that takes g O(g) n 3/2 arithmetic operations. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. For graphs of any fixed genus, our algorithms improve these time bounds by roughly a factor of n. Our key insight is to optimize the homology class of the flow, rather than directly optimizing the flow itself; two flows are in the same homology class if their difference is a weighted sum of directed facial cycles. A dual formulation of our algorithm computes the minimum-cost circulation in a given (real or integer) homology class.

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“…In the case of a non-orientable surface, the signature of an edge is also given, specifying if the orientation of the rotation switches across this edge. Since computing or approximating a low-genus embedding of a non-planar graph is an NP-complete problem [5,18], we require the embedding to be given as part of the input and we consider reachability in G(m, g) to be a promise problem. In the case of genus zero, we can compute a planar embedding in log-space and the promise condition can be removed.…”

Section: Topological Embeddings and Algorithmsmentioning

confidence: 99%

Space-Efficient Algorithms for Reachability in Surface-Embedded Graphs

Stolee

Vinodchandran

2012

2012 IEEE 27th Conference on Computational Complexity

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This work presents a log-space reduction which compresses a directed acyclic graph with m sources embedded on a surface of genus g to a graph on O(m + g) vertices while preserving reachability between a given pair of vertices. Applying existing algorithms to this smaller graph gives improved space bounds as well as improved simultaneous time-space bounds for the reachability problem for a large class of directed acyclic graphs. Specifically, it significantly extends the class of surface-embedded graphs with log-space reachability algorithms: from planar graphs with O(log n) sources, to graphs with 2 O( √ log n) sources embedded in a surface of genus 2 O( √ log n) . Additionally it also yields sublinear space (n 1− space) algorithms with polynomial running time for graphs with n 1− sources embedded on surfaces of genus n 1− .

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A note on approximating graph genus

Cited by 17 publications

References 9 publications

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Homology Flows, Cohomology Cuts

Space-Efficient Algorithms for Reachability in Surface-Embedded Graphs

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