TY - GEN

T1 - Optimal homologous cycles, total unimodularity, and linear programming

AU - Dey, Tamal K.

AU - Hirani, Anil N.

AU - Krishnamoorthy, Bala

PY - 2010

Y1 - 2010

N2 - 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 [∂p+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.

AB - 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 [∂p+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.

KW - computational topology

KW - homology localization

KW - optimal chain

KW - simplicial complex

UR - http://www.scopus.com/inward/record.url?scp=77954735165&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954735165&partnerID=8YFLogxK

U2 - 10.1145/1806689.1806721

DO - 10.1145/1806689.1806721

M3 - Conference contribution

AN - SCOPUS:77954735165

SN - 9781605588179

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 221

EP - 230

BT - STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing

T2 - 42nd ACM Symposium on Theory of Computing, STOC 2010

Y2 - 5 June 2010 through 8 June 2010

ER -