Chen, ChaoIST Austria ; Freedman, Daniel
We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We study the problem with different measures: volume, diameter and radius. For volume, that is, the 1-norm of a cycle, two main results are presented. First, we prove that 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). The latter result leads to 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. As for the other two measures defined by pairwise geodesic distance, diameter and radius, we show that the localization problem is NP-hard for diameter but is polynomial for radius. Our work is restricted to homology over the ℤ2 field.
Discrete & Computational Geometry
Austrian Science Fund grant P20134-N13
425 - 448
Chen C, Freedman D. Hardness results for homology localization. Discrete & Computational Geometry. 2011;45(3):425-448. doi:10.1007/s00454-010-9322-8
Chen, C., & Freedman, D. (2011). Hardness results for homology localization. Discrete & Computational Geometry, 45(3), 425–448. https://doi.org/10.1007/s00454-010-9322-8
Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” Discrete & Computational Geometry 45, no. 3 (2011): 425–48. https://doi.org/10.1007/s00454-010-9322-8.
C. Chen and D. Freedman, “Hardness results for homology localization,” Discrete & Computational Geometry, vol. 45, no. 3, pp. 425–448, 2011.
Chen C, Freedman D. 2011. Hardness results for homology localization. Discrete & Computational Geometry. 45(3), 425–448.
Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” Discrete & Computational Geometry, vol. 45, no. 3, Springer, 2011, pp. 425–48, doi:10.1007/s00454-010-9322-8.