Mathematics – Algebraic Topology
Scientific paper
2010-01-02
Mathematics
Algebraic Topology
Earlier version of this paper appeared in the 42nd ACM Symposium on Theory of Computing (STOC 2010). In this version we comple
Scientific paper
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 coefficients, we show the following : 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 $L_0$ is a subset of $L$. 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 NP-hard under $\mathbb{Z}_2$ 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 2-cycle in a given homology class for any finite simplicial complex embedded in $\mathbb{R}^3$. 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.
Dey Tamal K.
Hirani Anil N.
Krishnamoorthy Bala
No associations
LandOfFree
Optimal Homologous Cycles, Total Unimodularity, and Linear Programming does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Optimal Homologous Cycles, Total Unimodularity, and Linear Programming, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Optimal Homologous Cycles, Total Unimodularity, and Linear Programming will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-222667