Hardness Results for Homology Localization

: © Hardness Results for Homology Localization Chao Chen, Daniel Freedman HP Laboratories HPL-2009-374 algebraic topology, homology, localization We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1-norm of a cycle. Two main results are presented. First, we prove the problem is NP-hard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NP-hard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the Z2 field. External Posting Dat...

• Type: Journal
• Year: 2010
• Where: SODA
• Authors: Chao Chen, Daniel Freedman
• Keywords: Algorithms, Geometric Measures, Homology Localization, Homology Localization Chao, SODA 2010