A strongly polynomial minimum cost circulation algorithm
Combinatorica
A strongly polynomial algorithm to solve combinatorial linear programs
Operations Research
Theory of linear and integer programming
Theory of linear and integer programming
Introduction to Linear Optimization
Introduction to Linear Optimization
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Tightening non-simple paths and cycles on surfaces
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Computing geometry-aware handle and tunnel loops in 3D models
ACM SIGGRAPH 2008 papers
Splitting (complicated) surfaces is hard
Computational Geometry: Theory and Applications
Minimum cuts and shortest homologous cycles
Proceedings of the twenty-fifth annual symposium on Computational geometry
Measuring and computing natural generators for homology groups
Computational Geometry: Theory and Applications
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
The least spanning area of a knot and the optimal bounding chain problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
Minimum cuts and shortest non-separating cycles via homology covers
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
ACM Transactions on Mathematical Software (TOMS)
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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: For a finite simplicial complex K of dimension greater than p, the boundary matrix [partialp+1] is totally unimodular if and only if Hp(L, L0) is torsion-free, 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 an 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 NP-hard under Z2 coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal (d-1)-cycle in a given homology class for any triangulation of an orientable compact d-manifold or for any finite simplicial complex embedded in Rd. 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.