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Shortest Nontrivial Cycles in Directed and Undirected Surface Graphs
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest ..."
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Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest nonseparating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest noncontractible or nonnullhomologous cycle in 2O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest nontrivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest nonseparating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
Structured Recursive Separator Decompositions for Planar Graphs in Linear Time
, 2012
"... Given a planar graph G on n vertices and an integer parameter r < n, an r–division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for ea ..."
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Given a planar graph G on n vertices and an integer parameter r < n, an r–division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O( r). We provide a lineartime algorithm for computing r–divisions with few holes. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r–divisions for essentially all values of r. In particular, given an increasing sequence r = (r1, r2,...), our algorithm can produce a recursive r–division with few holes in linear time. r–divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our lineartime algorithm improves upon the decomposition algorithm used in the stateoftheart algorithm for minimum st–cut (Italiano, Nussbaum, Sankowski, and WulffNilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and boundedgenus graphs).
Faster shortest noncontractible cycles in directed surface graphs
 CoRR
"... Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. ..."
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Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest nonseparating cycle.
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
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We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual
Multicuts in Planar and BoundedGenus Graphs with Bounded Number of Terminals
, 2015
"... Given an undirected, edgeweighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimumweight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Rely ..."
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Given an undirected, edgeweighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimumweight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Relying on topological techniques, we provide a polynomialtime algorithm for this problem in the case where G is embedded on a fixed surface of genus g (e.g., when G is planar) and has a fixed number t of terminals. The running time is a polynomial
Counting and Sampling Minimum Cuts in Genus g Graphs
, 2012
"... Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm ..."
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Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are few prior results that are fixed parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count the number of cuts once, can sample repeatedly for a minimum cut in O(g 2 n) time per sample.
AllPairs Minimum Cuts in NearLinear Time for SurfaceEmbedded Graphs
"... For an undirected nvertex graph G with nonnegative edgeweights, we consider the following type of query: given two vertices s and t in G, what is the weight of a minimum stcut in G? We solve this problem in preprocessing time O(n log3 n) for graphs of bounded genus, giving the first subquadrati ..."
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For an undirected nvertex graph G with nonnegative edgeweights, we consider the following type of query: given two vertices s and t in G, what is the weight of a minimum stcut in G? We solve this problem in preprocessing time O(n log3 n) for graphs of bounded genus, giving the first subquadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and WulffNilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory–Hu tree for the given graph, providing a data structure with space O(n) that can answer minimumcut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is 2O(g 2).
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
, 2014
"... How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically nontrivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (o ..."
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How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically nontrivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual crossmetric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of nontrivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete setting, and viceversa. This implies a conjecture by Przytycka and Przytycki from 1993, a number of new systolic inequalities in the discrete setting, and the fact that a theorem of Hutchinson on the edgewidth of triangulated surfaces and Gromov’s systolic inequality for surfaces are essentially equivalent. We also discuss how these proofs generalize to higher dimensions. Then we focus on topological decompositions of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions of length O(g3/2n1/2) for any triangulated combinatorial surface of genus g with n triangles, and describe an O(gn)time algorithm to compute such a decomposition. Finally, we consider the problem of embedding a cut graph (or more generally a cellular graph) with a given combinatorial map on a given surface. Using random triangulations, we prove (essentially) that, for any choice of a combinatorial map, there are some surfaces on which any cellular embedding with that combinatorial map has length superlinear in the number of triangles of the triangulated combinatorial surface. There is also a similar result for graphs embedded on polyhedral triangulations.