Results 1  10
of
17
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 ..."
Abstract

Cited by 30 (10 self)
 Add to MetaCart
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.
Optimal Homologous Cycles, Total Unimodularity, and Linear Programming
, 2010
"... Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following (Theorem 5.2): For a fin ..."
Abstract

Cited by 29 (11 self)
 Add to MetaCart
Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer (Z) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex K of dimension greater than p, the boundary matrix [∂p+1] is totally unimodular if and only if Hp(L, L0) is torsionfree, for all pure subcomplexes L0, L in K of dimensions p and p + 1 respectively, where L0 ⊂ L. Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain integer solution. Thus the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NPhard under Z2 coefficients which, being a field, is in general easier to deal with. One consequence of our result, among others, is that one can compute in polynomial time an optimal 2cycle in a given homology class for any finite simplicial complex embedded in R 3. Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles.
Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 SYMPOSIUM ON DISCRETE ALGORITHMS
, 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest nonseparating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2homology cover.
The least spanning area of a knot and the optimal bounding chain problem
 In Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG’11). ACM
, 2011
"... Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, th ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3dimensional manifold. When the knot is embedded in a general 3manifold, the problems of finding these surfaces were shown to be NPcomplete and NPhard respectively. However, there is evidence that the special case when the ambient manifold is R3, or more generally when the second homology is trivial, should be considerably more tractable. Indeed, we show here that a natural discrete version of the least area surface can be found in polynomial time. The precise setting is that the knot is a 1dimensional subcomplex of a triangulation of the ambient 3manifold. The main tool we use is a linear programming formulation of the Optimal Bounding Chain Problem (OBCP), where one is required to find the smallest norm chain with a given boundary. While the decision variant of OBCP is NPcomplete in general, we give conditions under which it can be solved in polynomial time. We then show that the least area surface can be constructed from the optimal bounding chain using a standard desingularization argument from 3dimensional topology. We also prove that the related Optimal Homologous Chain Problem is NPcomplete for homology with integer coefficients, complementing the corresponding result of Chen and Freedman for mod 2 homology.
Approximating loops in a shortest homology basis from point data
 arXiv:0909.5654v2[cs.CG] (2009), Online. URL http://arxiv.org/abs/0909.5654
"... Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manif ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of loops from a point data that presumably sample a smooth manifold M ⊂ R d. These loops approximate a shortest basis of the one dimensional homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1(K) for any finite simplicial complex K whose edges have nonnegative weights.
Global Minimum Cuts in Surface Embedded Graphs
"... We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm kno ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We give a deterministic algorithm to find the minimum cut in a surfaceembedded graph in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log³ n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n² log n) time. We can also achieve a deterministic g O(g) n² log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.
Annotating simplices with a homology basis and its applications
 In Proc. 13th Scandinavian Symp. and Workshop on Algorithm Theory
, 2012
"... Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in an ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
Let K be a simplicial complex and g the rank of its pth homology group Hp(K) defined with Z2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each psimplex of K with a binary vector of length g with the following property: the annotations, summed over all psimplices in any pcycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω) time, where n is the size of K and ω < 2.376 is a quantity so that two n × n matrices can be multiplied in O(n ω) time. The precomputation of annotations permits answering queries about the independence or the triviality of pcycles efficiently. Using annotations of edges in 2complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1dimensional homology. Specifically, for computing an optimal basis of H1(K), we improve the time complexity known for the problem from O(n 4) to O(n ω + n 2 g ω−1). Here n denotes the size of the 2skeleton of K and g the rank of H1(K). Computing an optimal cycle homologous to a given 1cycle is NPhard even for surfaces and an algorithm taking 2 O(g) n log n time is known for surfaces. We extend this algorithm to work with arbitrary 2complexes in O(n ω) + 2 O(g) n 2 log n time using annotations. 1
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.
Approximating cycles in a shortest basis of the first homology group from point data. Inverse Problems
, 2012
"... Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of cycles from a point data that presumably sample a smooth mani ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
(Show Context)
Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper we present an algorithm to compute a set of cycles from a point data that presumably sample a smooth manifold M ⊂ R d. These cycles approximate a shortest basis of the first homology group H1(M) over coefficients in finite field Z2. Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving our result, we also present a polynomial time algorithm for computing a shortest basis of H1(K) for any finite simplicial complex K whose edges have nonnegative weights.