In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch [15]. For D-dimensional simplicial complexes, we obtain a (D+1) /(D 2 +D+1)-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. We also describe a 2 /D-factor approximation algorithm for simplicial manifolds by processing the simplices in increasing order of dimension. This algorithm may also be applied to non-manifolds resulting in a 1 /(D+1)-factor approximation ratio. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results. * abhishek@jcrathod.in † tbmasood@csa.iisc.ernet.in ‡ vijayn@csa.iisc.ernet.in

Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant of an input simplicial complex to computing the same topological invariant of a (significantly smaller) collapsed cell or chain complex. Consequently, devising methods for obtaining gradient vector fields on complexes to reduce the size of the problem instance has become an emerging theme over the last decade. While computing the optimal gradient vector field on a simplicial complex is NP-hard, several heuristics have been observed to compute near-optimal gradient vector fields on a wide variety of datasets. Understanding the theoretical limits of these strategies is therefore a fundamental problem in computational topology.In this paper, we consider the approximability of maximization and minimization variants of the Morse matching problem. We establish hardness results for Max-Morse matching and Min-Morse matching, settling an open problem posed by Joswig and Pfetsch [19]. In particular, we show that, for a simplicial complex of dimension d ≥ 3 with n simplices, it is NP-hard to approximate Min-Morse matching within a factor of O(n 1−ǫ ), for any ǫ > 0. Moreover, we establish hardness of approximation results for Max-Morse matching for simplicial complexes of dimension d ≥ 2, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching.

Sterilization in Orthodontics and particularly in the entire dental practice is an important topic which requires special attention because both the patients and the practitioners have a substantial risk of spreading infections like hepatitis B, pneumonia, tuberculosis and HIV. Control of infection that spreads through various instruments and armamentarium used in the field of orthodontics and also in the dental practice is of utmost importance to prevent cross-infection. The present article reviews various recent methods of sterilization for an effective and efficient infection-free orthodontic practice.

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In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes. We propose two novel variants of the standard algorithm, called swap and retrospective reductions, which improve upon state-of-the-art techniques on several examples in practice. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worstcase complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

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