计算最小持续循环：多项式和困难情况
Computing Minimal Persistent Cycles: Polynomial and Hard Cases
摘要：
持久性循环，特别是最小循环，是有用的几何特征，在纯拓扑持久性图（也称为条形码）中用作区间的扩充。在我们早期的工作中，我们证明了计算有限区间的最小1维持久圈（持久1-圈）是NP-困难的，而计算无限区间的最小1维持久圈（持久1-圈）是多项式易处理的。在本文中，我们解决这个问题的一般尺寸与$\mathbb{Z}_2$系数。除了证明它是NP-困难的计算最小持久d-圈（d>1）的任意单纯复形的两种类型的间隔，我们确定了两个有趣的情况下，这是多项式听话。这两种情况假设复形是流形的某种推广，我们称之为弱伪流形。对于弱（d+1）-伪流形的d次持久图的有限区间，利用了这样的区间的持久圈是零同调的这一事实，将问题归结为最小割问题.由于无穷区间上的问题是NP-难的，我们进一步假设弱（d+1）-伪流形嵌入到$\mathbb{R}^{d+1}$中，使得复形具有自然的对偶图结构，问题转化为最小割问题.这两种算法在科学数据上的实验表明，最小持久周期捕获数据的各种重要特征。
英文摘要：
Persistent cycles, especially the minimal ones, are useful geometric features functioning as augmentations for the intervals in a purely topological persistence diagram (also termed as barcode). In our earlier work, we showed that computing minimal 1-dimensional persistent cycles (persistent 1-cycles) for finite intervals is NP-hard while the same for infinite intervals is polynomially tractable. In this paper, we address this problem for general dimensions with $\mathbb{Z}_2$ coefficients. In addition to proving that it is NP-hard to compute minimal persistent d-cycles (d>1) for both types of intervals given arbitrary simplicial complexes, we identify two interesting cases which are polynomially tractable. These two cases assume the complex to be a certain generalization of manifolds which we term as weak pseudomanifolds. For finite intervals from the d-th persistence diagram of a weak (d+1)-pseudomanifold, we utilize the fact that persistent cycles of such intervals are null-homologous and reduce the problem to a minimal cut problem. Since the same problem for infinite intervals is NP-hard, we further assume the weak (d+1)-pseudomanifold to be embedded in $\mathbb{R}^{d+1}$ so that the complex has a natural dual graph structure and the problem reduces to a minimal cut problem. Experiments with both algorithms on scientific data indicate that the minimal persistent cycles capture various significant features of the data.
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