arXiv subjects

Bala Krishnamoorthy

Publications and source records attributed to Bala Krishnamoorthy.

39 records · Page 3Linked to original sources

arXiv2013

Edge Contractions and Simplicial Homology

We study the effect of edge contractions on simplicial homology because these contractions have turned to be useful in various applications involving topology. It was observed previously that contracting edges that satisfy the so called link condition preserves homeomorphism in low dimensional complexes, and homotopy in general. But, checking the link condition involves computation in all dimensions, and hence can be costly, especially in high dimensional complexes. We define a weaker and more local condition called the p-link condition for each dimension p, and study its effect on edge contractions. We prove the following: (i) For homology groups, edges satisfying the p- and (p-1)-link conditions can be contracted without disturbing the p-dimensional homology group. (ii) For relative homology groups, the (p-1)-, and the (p-2)-link conditions suffice to guarantee that the contraction does not introduce any new class in any of the resulting relative homology groups, though some of the existing classes can be destroyed. Unfortunately, the surjection in relative homolgy groups does not guarantee that no new relative torsion is created. (iii) For torsions, edges satisfying the p-link condition alone can be contracted without creating any new relative torsion and the p-link condition cannot be avoided. The results on relative homology and relative torsion are motivated by recent results on computing optimal homologous chains, which state that such problems can be solved by linear programming if the complex has no relative torsion. Edge contractions that do not introduce new relative torsions, can safely be availed in these contexts.

cs.CG↗

arXiv2011

Optimal Homologous Cycles, Total Unimodularity, and Linear Programming

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.

math.AT↗

arXiv2009

Column basis reduction, and decomposable knapsack problems

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b' <= Ax <= b, x \in Z^n with b' <= AUy <= b, y \in Z^n, where U is a unimodular matrix computed via basis reduction, to make the columns of $AU$ short and nearly orthogonal. The reformulation is called rangespace reformulation. It is motivated by the reformulation technique proposed for equality constrained IPs by Aardal, Hurkens and Lenstra. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, Chvatal and Todd, Avis, Aardal and Lenstra, and Cornuejols et al. DKPs are knapsack problems with a constraint vector of the form $pM + r, $ with $p >0$ and $r$ integral vectors, and $M$ a large integer. If the parameters are suitably chosen in DKPs, we prove 1) hardness results for these problems, when branch-and-bound branching on individual variables is applied; 2) that they are easy, if one branches on the constraint $px$ instead; and 3) that branching on the last few variables in either the rangespace- or the AHL-reformulations is equivalent to branching on $px$ in the original problem. We also provide recipes to generate such instances. Our computational study confirms that the behavior of the studied instances in practice is as predicted by the theoretical results.

math.OC↗