Document

# Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes

## File

LIPIcs.SoCG.2020.21.pdf
• Filesize: 0.51 MB
• 15 pages

## Cite As

Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 21:1-21:15, Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.21

## Abstract

We study two optimization problems on simplicial complexes with homology over β€β, the minimum bounded chain problem: given a d-dimensional complex π¦ embedded in β^(d+1) and a null-homologous (d-1)-cycle C in π¦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold β³ and a d-chain D in β³, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(β{log Ξ²_d})-approximation algorithm for the minimum bounded chain problem where Ξ²_d is the dth Betti number of π¦. Finally, we provide an O(β{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in β³.

## Subject Classification

##### ACM Subject Classification
• Theory of computation β Computational geometry
##### Keywords
• computational topology
• algorithmic complexity
• simplicial complexes

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(βlog n) approximation algorithms for min uncut, min 2CNF deletion, and directed cut problems. In Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 573-581, New York, NY, USA, 2005. ACM. URL: https://doi.org/10.1145/1060590.1060675.
2. Ian Agol, Joel Hass, and William Thurston. The computational complexity of knot genus and spanning area. Transactions of the American Mathematical Society, 358(09):3821-3851, September 2006. URL: https://doi.org/10.1090/s0002-9947-05-03919-x.
3. Per Austrin and Subhash Khot. A simple deterministic reduction for the gap minimum distance of code problem. In Proceedings of the 38th International Colloquim Conference on Automata, Languages and Programming - Volume Part I, ICALP'11, pages 474-485, Berlin, Heidelberg, 2011. Springer-Verlag.
4. E. Berlekamp, R. McEliece, and H. van Tilborg. On the inherent intractability of certain coding problems (corresp.). IEEE Transactions on Information Theory, 24(3):384-386, May 1978. URL: https://doi.org/10.1109/TIT.1978.1055873.
5. Arnab Bhattacharyya, Ameet Gadekar, Suprovat Ghoshal, and Rishi Saket. On the hardness of learning sparse parities. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 11:1-11:17, 2016.
6. Oleksiy Busaryev, Sergio Cabello, Chao Chen, Tamal K. Dey, and Yusu Wang. Annotating simplices with a homology basis and its applications. In Fedor V. Fomin and Petteri Kaski, editors, Algorithm Theory - SWAT 2012, pages 189-200, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
7. J. Carvalho, Mikael Vejdemo-Johansson, Danica Kragic, and Florian Pokorny. An algorithm for calculating top-dimensional bounding chains. PeerJ Computer Science, 4:e153, May 2018. URL: https://doi.org/10.7717/peerj-cs.153.
8. Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Minimum cuts and shortest homologous cycles. In Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, SCG '09, pages 377-385, New York, NY, USA, 2009. ACM. URL: https://doi.org/10.1145/1542362.1542426.
9. Erin Wolf Chambers and Mikael Vejdemo-Johansson. Computing minimum area homologies. Comput. Graph. Forum, 34(6):13β21, September 2015. URL: https://doi.org/10.1111/cgf.12514.
10. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Comput. Complex., 15(2):94-114, June 2006. URL: https://doi.org/10.1007/s00037-006-0210-9.
11. Chao Chen and Daniel Freedman. Hardness results for homology localization. Discrete & Computational Geometry, 45(3):425-448, April 2011. URL: https://doi.org/10.1007/s00454-010-9322-8.
12. A.V. Chernavsky and V.P. Leksine. Unrecognizability of manifolds. Annals of Pure and Applied Logic, 141(3):325-335, 2006. Papers presented at the Second St. Petersburg Days of Logic and Computability Conference on the occasion of the centennial of Andrey Andreevich Markov, Jr. URL: https://doi.org/10.1016/j.apal.2005.12.011.
13. Tamal K. Dey, Anil N. Hirani, and Bala Krishnamoorthy. Optimal homologous cycles, total unimodularity, and linear programming. SIAM J. Comput., 40(4):1026-1044, July 2011. URL: https://doi.org/10.1137/100800245.
14. Tamal K. Dey, Tao Hou, and Sayan Mandal. Computing minimal persistent cycles: Polynomial and hard cases. ArXiv, abs/1907.04889, 2019. URL: http://arxiv.org/abs/1907.04889.
15. Rod G. Downey, Michael R. Fellows, Alexander Vardy, and Geoff Whittle. The parametrized complexity of some fundamental problems in coding theory. SIAM J. Comput., 29(2):545-570, October 1999. URL: https://doi.org/10.1137/S0097539797323571.
16. Nathan M. Dunfield and Anil N. Hirani. The least spanning area of a knot and the optimal bounding chain problem. In Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG '11, pages 135-144, New York, NY, USA, 2011. ACM. URL: https://doi.org/10.1145/1998196.1998218.
17. Jeff Erickson and Amir Nayyeri. Minimum cuts and shortest non-separating cycles via homology covers. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11, pages 1166-1176, Philadelphia, PA, USA, 2011. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2133036.2133124.
18. Leo Grady. Minimal surfaces extend shortest path segmentation methods to 3D. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32:321-334, 2010.
19. Joshua A. Grochow and Jamie Tucker-Foltz. Computational Topology and the Unique Games Conjecture. In Bettina Speckmann and Csaba D. TΓ³th, editors, 34th International Symposium on Computational Geometry (SoCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages 43:1-43:16, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.43.
20. Allen Hatcher. Algebraic topology. Cambridge University Press, 2002.
21. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM J. Comput., 37(1):319-357, April 2007. URL: https://doi.org/10.1137/S0097539705447372.
22. Subhash A. Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into πβ. J. ACM, 62(1):8:1-8:39, March 2015. URL: https://doi.org/10.1145/2629614.
23. Danil Kirsanov and Steven Gortler. A discrete global minimization algorithm for continuous variational problems, August 2004.
24. Pasin Manurangsi and Luca Trevisan. Mildly exponential time approximation algorithms for vertex cover, balanced separator and uniform sparsest cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, pages 20:1-20:17, 2018. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.20.
25. James R. Munkres. Elements of Algebraic Topology. Addison Wesley Publishing Company, 1984. URL: http://www.worldcat.org/isbn/0201045869.
26. Christos Papadimitriou and Mihalis Yannakakis. Optimization, approximation, and complexity classes. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC '88, pages 229-234, New York, NY, USA, 1988. ACM. URL: https://doi.org/10.1145/62212.62233.
27. Igor Razgon and Barry O'Sullivan. Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, December 2009. URL: https://doi.org/10.1016/j.jcss.2009.04.002.
28. John M. Sullivan. A Crystalline Approximation Theorem for Hypersurfaces. PhD thesis, Princeton University, 1990. URL: http://torus.math.uiuc.edu/jms/Papers/thesis.
29. A. Vardy. The intractability of computing the minimum distance of a code. IEEE Trans. Inf. Theor., 43(6):1757-1766, November 1997.