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Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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Cited by 16 (6 self)
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. 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 surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
Multiplesource shortest paths in embedded graphs
, 2012
"... Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of ..."
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Cited by 8 (5 self)
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Let G be a directed graph with n vertices and nonnegative weights in its directed edges, embedded on a surface of genus g, and let f be an arbitrary face of G. We describe an algorithm to preprocess the graph in O(gn log n) time, so that the shortestpath distance from any vertex on the boundary of f to any other vertex in G can be retrieved in O(log n) time. Our result directly generalizes the O(n log n)time algorithm of Klein [Multiplesource shortest paths in planar graphs. In Proc. 16th Ann. ACMSIAM Symp. Discrete Algorithms, 2005] for multiplesource shortest paths in planar graphs. Intuitively, our preprocessing algorithm maintains a shortestpath tree as its source point moves continuously around the boundary of f. As an application of our algorithm, we describe algorithms to compute a shortest noncontractible or nonseparating cycle in embedded, undirected graphs in O(g² n log n) time.
PolynomialTime Exact Inference in NPHard Binary MRFs via Reweighted Perfect Matching
 13TH INTL. CONF. ARTIFICIAL INTELLIGENCE AND STATISTICS (AISTATS
, 2010
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Planar Drawings of HigherGenus Graphs
"... Abstract. In this paper, we give polynomialtime algorithms that can take a graph G with a given combinatorial embedding on a surface S of genus g and produce a planar drawing of G in R 2, with a bounding face defined by a polygonal schema P for S. Our drawings are planar, but they allow for multipl ..."
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Cited by 4 (1 self)
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Abstract. In this paper, we give polynomialtime algorithms that can take a graph G with a given combinatorial embedding on a surface S of genus g and produce a planar drawing of G in R 2, with a bounding face defined by a polygonal schema P for S. Our drawings are planar, but they allow for multiple copies of vertices and edges on P’s boundary, which is a common way of visualizing highergenus graphs in the plane. As a side note, we show that it is NPcomplete to determine whether a given graph embedded in a genusg surface has a set of 2g fundamental cycles with vertexdisjoint interiors, which would be desirable from a graphdrawing perspective. 1
Genus characterizes the complexity of certain graph problems: Some tight results
 pp.892–907 in Journal of Computer and System Sciences vol.73:6
, 2007
"... We study the fixedparameter tractability, subexponential time computability, and approximability of the wellknown NPhard problems: independent set, vertex cover, and dominating set. We derive tight results and show that the computational complexity of these problems, with respect to the above com ..."
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Cited by 1 (0 self)
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We study the fixedparameter tractability, subexponential time computability, and approximability of the wellknown NPhard problems: independent set, vertex cover, and dominating set. We derive tight results and show that the computational complexity of these problems, with respect to the above complexity measures, is dependent on the genus of the underlying graph. For instance, we show that, under the widelybelieved complexity assumption W [1] ̸ = FPT, independent set on graphs of genus bounded by g1(n) is fixed parameter tractable if and only if g1(n) = o(n2), and dominating set on graphs of genus bounded by g2(n) is fixed parameter tractable if and only if g2(n) = no(1). Under the assumption that not all SNP problems are solvable in subexponential time, we show that the above three problems on graphs of genus bounded by g3(n) are solvable in subexponential time if and only if g3(n) = o(n). We also show that the independent set, the kernelized vertex cover, and the kernelized dominating set problems on graphs of genus bounded by g4(n) have PTAS if g4(n) = o(n / log n), and that, under the assumption P ̸ = NP, the independent set problem on graphs of genus bounded by g5(n) has no PTAS if g5(n) = Ω(n), and the vertex cover and dominating set problems on graphs of genus bounded by g6(n) have no PTAS if g6(n) = nΩ(1). 1