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
A decomposition theory for matroids VII: analysis of minimal violation matrices
Journal of Combinatorial Theory Series B
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
Hardness results for homology localization
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Global minimum cuts in surface embedded graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Annotating simplices with a homology basis and its applications
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Enumerating colorings, tensions and flows in cell complexes
Journal of Combinatorial Theory Series A
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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 ($\mathbb{Z}$) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex $K$ of dimension greater than $p$, the boundary matrix $[\partial_{p+1}]$ is totally unimodular if and only if $H_p(L, L_0)$ is torsion-free for all pure subcomplexes $L_0, L$ in $K$ of dimensions $p$ and $p+1$, respectively, where $L_0 \subsetL$. 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 $\mathbb{Z}_2$ 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 $\mathbb{R}^d$. 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. Our result can also be viewed as providing a topological characterization of total unimodularity.