Efficient Algorithms for Computing a Minimal Homology Basis

Abstract

Efficient computation of shortest cycles which form a homology basis under \(\mathbb {Z}_2\)-additions in a given simplicial complex \(\mathcal {K}\) has been researched actively in recent years. When the complex \(\mathcal {K}\) is a weighted graph with n vertices and m edges, the problem of computing a shortest (homology) cycle basis is known to be solvable in \(O(m^2n/\log n+ n^2m)\)-time. Several works [1, 2] have addressed the case when the complex \(\mathcal {K}\) is a 2-manifold. The complexity of these algorithms depends on the rank g of the one-dimensional homology group of \(\mathcal {K}\). This rank g has a lower bound of \(\varTheta (n)\), where n denotes the number of simplices in \(\mathcal {K}\), giving an \(O(n^4)\) worst-case time complexity for the algorithms in [1, 2]. This worst-case complexity is improved in [3] to \(O(n^\omega + n^2g^{\omega -1})\) for general simplicial complexes where \(\omega < 2.3728639\) [4] is the matrix multiplication exponent. Taking \(g=\varTheta (n)\), this provides an \(O(n^{\omega +1})\) worst-case algorithm. In this paper, we improve this time complexity. Combining the divide and conquer technique from [5] with the use of annotations from [3], we present an algorithm that runs in \(O(n^\omega +n^2g)\) time giving the first \(O(n^3)\) worst-case algorithm for general complexes. If instead of minimal basis, we settle for an approximate basis, we can improve the running time even further. We show that a 2-approximate minimal homology basis can be computed in \(O(n^{\omega }\sqrt{n \log n})\) expected time. We also study more general measures for defining the minimal basis and identify reasonable conditions on these measures that allow computing a minimal basis efficiently.