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Finding shortest nontrivial cycles in directed graphs on surfaces
 In These Proceedings
, 2010
"... Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected ..."
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Let D be a weighted directed graph cellularly embedded in a surface of genus g, orientable or not, possibly with boundary. We describe algorithms to compute a shortest noncontractible and a shortest surface nonseparating cycle in D. This generalizes previous results that only dealt with undirected graphs. Our first algorithm computes such cycles in O(n 2 log n) time, where n is the total number of vertices and edges of D, thus matching the complexity of the best known algorithm in the undirected case. It revisits and extends Thomassen’s 3path condition; the technique applies to other families of cycles as well. We also give an algorithm with subquadratic complexity in the complexity of the input graph, if g is fixed. Specifically, we can solve the problem in O ( √ g n 3/2 log n) time, using a divideandconquer technique that simplifies the graph while preserving the topological properties of its cycles. A variant runs in O(ng log g + nlog 2 n) for graphs of bounded treewidth.
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
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.
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.
COMBINATORIAL OPTIMIZATION ON EMBEDDED CURVES
, 2012
"... We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compu ..."
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We describe several algorithms for classifying, comparing and optimizing curves on surfaces. We give algorithms to compute the minimum member of a given homology class, particularly computing the maximum flow and minimum cuts, in surface embedded graphs. We describe approximation algorithms to compute certain similarity measures for embedded curves on a surface. Finally, we present algorithms to solve computational problems for compactly presented curves. We describe the first algorithms to compute the shortest representative of a Z2homology class. Given a directed graph embedded on a surface of genus g with b boundary cycles, we can compute the shortest single cycle Z2homologous to a given even subgraph in 2O(g+b)n log n time. As a consequence we obtain an algorithm to compute the shortest directed nonseparating cycle in 2O(g)n log n time, which improves the previous best algorithm by a factor of O( p n) if the genus is