---
_id: '10909'
abstract:
- lang: eng
text: 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.
acknowledgement: Partially supported by the Austrian Science Fund under grantFSP-S9103-N04
and P20134-N13.
article_processing_charge: No
author:
- first_name: Chao
full_name: Chen, Chao
id: 3E92416E-F248-11E8-B48F-1D18A9856A87
last_name: Chen
- first_name: Daniel
full_name: Freedman, Daniel
last_name: Freedman
citation:
ama: 'Chen C, Freedman D. Hardness results for homology localization. In: Proceedings
of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms. Society for
Industrial and Applied Mathematics; 2010:1594-1604. doi:10.1137/1.9781611973075.129'
apa: 'Chen, C., & Freedman, D. (2010). Hardness results for homology localization.
In Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms
(pp. 1594–1604). Austin, TX, United States: Society for Industrial and Applied
Mathematics. https://doi.org/10.1137/1.9781611973075.129'
chicago: Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.”
In Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms,
1594–1604. Society for Industrial and Applied Mathematics, 2010. https://doi.org/10.1137/1.9781611973075.129.
ieee: C. Chen and D. Freedman, “Hardness results for homology localization,” in
Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms,
Austin, TX, United States, 2010, pp. 1594–1604.
ista: 'Chen C, Freedman D. 2010. Hardness results for homology localization. Proceedings
of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium
on Discrete Algorithms, 1594–1604.'
mla: Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.”
Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms,
Society for Industrial and Applied Mathematics, 2010, pp. 1594–604, doi:10.1137/1.9781611973075.129.
short: C. Chen, D. Freedman, in:, Proceedings of the 2010 Annual ACM-SIAM Symposium
on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2010,
pp. 1594–1604.
conference:
end_date: 2010-01-19
location: Austin, TX, United States
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2010-01-17
date_created: 2022-03-21T08:24:07Z
date_published: 2010-02-01T00:00:00Z
date_updated: 2023-02-23T11:19:46Z
day: '01'
department:
- _id: HeEd
doi: 10.1137/1.9781611973075.129
language:
- iso: eng
month: '02'
oa_version: None
page: 1594-1604
publication: Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
eisbn:
- '9781611973075'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '3267'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Hardness results for homology localization
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2010'
...