---
_id: '1073'
abstract:
- lang: eng
text: Let X and Y be finite simplicial sets (e.g. finite simplicial complexes),
both equipped with a free simplicial action of a finite group G. Assuming that
Y is d-connected and dimX≤2d, for some d≥1, we provide an algorithm that computes
the set of all equivariant homotopy classes of equivariant continuous maps |X|→|Y|;
the existence of such a map can be decided even for dimX≤2d+1. This yields the
first algorithm for deciding topological embeddability of a k-dimensional finite
simplicial complex into Rn under the condition k≤23n−1. More generally, we present
an algorithm that, given a lifting-extension problem satisfying an appropriate
stability assumption, computes the set of all homotopy classes of solutions. This
result is new even in the non-equivariant situation.
author:
- first_name: Martin
full_name: Čadek, Martin
last_name: Čadek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Lukáš
full_name: Vokřínek, Lukáš
last_name: Vokřínek
citation:
ama: Čadek M, Krcál M, Vokřínek L. Algorithmic solvability of the lifting extension
problem. *Discrete & Computational Geometry*. 2017;54(4):915-965. doi:10.1007/s00454-016-9855-6
apa: Čadek, M., Krcál, M., & Vokřínek, L. (2017). Algorithmic solvability of
the lifting extension problem. *Discrete & Computational Geometry*, *54*(4),
915–965. https://doi.org/10.1007/s00454-016-9855-6
chicago: 'Čadek, Martin, Marek Krcál, and Lukáš Vokřínek. “Algorithmic Solvability
of the Lifting Extension Problem.” *Discrete & Computational Geometry*
54, no. 4 (2017): 915–65. https://doi.org/10.1007/s00454-016-9855-6.'
ieee: M. Čadek, M. Krcál, and L. Vokřínek, “Algorithmic solvability of the lifting
extension problem,” *Discrete & Computational Geometry*, vol. 54, no.
4, pp. 915–965, 2017.
ista: Čadek M, Krcál M, Vokřínek L. 2017. Algorithmic solvability of the lifting
extension problem. Discrete & Computational Geometry. 54(4), 915–965.
mla: Čadek, Martin, et al. “Algorithmic Solvability of the Lifting Extension Problem.”
*Discrete & Computational Geometry*, vol. 54, no. 4, Springer, 2017,
pp. 915–65, doi:10.1007/s00454-016-9855-6.
short: M. Čadek, M. Krcál, L. Vokřínek, Discrete & Computational Geometry 54
(2017) 915–965.
date_created: 2018-12-11T11:50:00Z
date_published: 2017-06-01T00:00:00Z
date_updated: 2019-08-02T12:36:47Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-016-9855-6
intvolume: ' 54'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1307.6444
month: '06'
oa: 1
oa_version: Submitted Version
page: 915 - 965
publication: Discrete & Computational Geometry
publication_identifier:
issn:
- '01795376'
publication_status: published
publisher: Springer
publist_id: '6309'
quality_controlled: '1'
status: public
title: Algorithmic solvability of the lifting extension problem
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 54
year: '2017'
...
---
_id: '568'
abstract:
- lang: eng
text: 'We study robust properties of zero sets of continuous maps f: X → ℝn. Formally,
we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets
of all continuous maps g closer to f than r in the max-norm. All of these sets
are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined
by A and an element of a certain cohomotopy group which (by a recent result) is
computable whenever the dimension of X is at most 2n - 3. By considering all r
> 0 simultaneously, the pointed cohomotopy groups form a persistence module-a
structure leading to persistence diagrams as in the case of persistent homology
or well groups. Eventually, we get a descriptor of persistent robust properties
of zero sets that has better descriptive power (Theorem A) and better computability
status (Theorem B) than the established well diagrams. Moreover, if we endow every
point of each zero set with gradients of the perturbation, the robust description
of the zero sets by elements of cohomotopy groups is in some sense the best possible
(Theorem C).'
author:
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
citation:
ama: Franek P, Krcál M. Persistence of zero sets. *Homology, Homotopy and Applications*.
2017;19(2):313-342. doi:10.4310/HHA.2017.v19.n2.a16
apa: Franek, P., & Krcál, M. (2017). Persistence of zero sets. *Homology,
Homotopy and Applications*, *19*(2), 313–342. https://doi.org/10.4310/HHA.2017.v19.n2.a16
chicago: 'Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” *Homology,
Homotopy and Applications* 19, no. 2 (2017): 313–42. https://doi.org/10.4310/HHA.2017.v19.n2.a16.'
ieee: P. Franek and M. Krcál, “Persistence of zero sets,” *Homology, Homotopy
and Applications*, vol. 19, no. 2, pp. 313–342, 2017.
ista: Franek P, Krcál M. 2017. Persistence of zero sets. Homology, Homotopy and
Applications. 19(2), 313–342.
mla: Franek, Peter, and Marek Krcál. “Persistence of Zero Sets.” *Homology, Homotopy
and Applications*, vol. 19, no. 2, International Press, 2017, pp. 313–42, doi:10.4310/HHA.2017.v19.n2.a16.
short: P. Franek, M. Krcál, Homology, Homotopy and Applications 19 (2017) 313–342.
date_created: 2018-12-11T11:47:14Z
date_published: 2017-01-01T00:00:00Z
date_updated: 2019-08-02T12:39:03Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.4310/HHA.2017.v19.n2.a16
intvolume: ' 19'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1507.04310
month: '01'
oa: 1
oa_version: Submitted Version
page: 313 - 342
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 2590DB08-B435-11E9-9278-68D0E5697425
grant_number: '701309'
name: Atomic-Resolution Structures of Mitochondrial Respiratory Chain Supercomplexes
(H2020)
publication: Homology, Homotopy and Applications
publication_identifier:
issn:
- '15320073'
publication_status: published
publisher: International Press
publist_id: '7246'
quality_controlled: '1'
status: public
title: Persistence of zero sets
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2017'
...
---
_id: '1237'
abstract:
- lang: eng
text: 'Bitmap images of arbitrary dimension may be formally perceived as unions
of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology
and homology groups are well known topological invariants of such sets. Cohomological
operations, such as the cup product, provide higher-order algebraic topological
invariants, especially important for digital images of dimension higher than 3.
If such an operation is determined at the level of simplicial chains [see e.g.
González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively
computable. However, decomposing a cubical complex into a simplicial one deleteriously
affects the efficiency of such an approach. In order to avoid this overhead, a
direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015,
253-275] for the cup product in cohomology, and implemented in the ChainCon software
package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for
the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series,
1947, 290-320] directly at the level of cubical chains, and we prove the correctness
of this formula. An implementation of this formula is programmed in C++ within
the ChainCon software framework. We provide a few examples and discuss the effectiveness
of this approach. One specific application follows from the fact that Steenrod
squares yield tests for the topological extension problem: Can a given map A →
Sd to a sphere Sd be extended to a given super-complex X of A? In particular,
the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value
r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the
extension problem.'
acknowledgement: The research conducted by both authors has received funding from
the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme (FP7/2007-2013) under REA grant agreements no. 291734 (for M. K.) and
no. 622033 (for P. P.).
alternative_title:
- LNCS
author:
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: 'Krcál M, Pilarczyk P. Computation of cubical Steenrod squares. In: Vol 9667.
Springer; 2016:140-151. doi:10.1007/978-3-319-39441-1_13'
apa: 'Krcál, M., & Pilarczyk, P. (2016). Computation of cubical Steenrod squares
(Vol. 9667, pp. 140–151). Presented at the CTIC: Computational Topology in Image
Context, Marseille, France: Springer. https://doi.org/10.1007/978-3-319-39441-1_13'
chicago: Krcál, Marek, and Pawel Pilarczyk. “Computation of Cubical Steenrod Squares,”
9667:140–51. Springer, 2016. https://doi.org/10.1007/978-3-319-39441-1_13.
ieee: 'M. Krcál and P. Pilarczyk, “Computation of cubical Steenrod squares,” presented
at the CTIC: Computational Topology in Image Context, Marseille, France, 2016,
vol. 9667, pp. 140–151.'
ista: 'Krcál M, Pilarczyk P. 2016. Computation of cubical Steenrod squares. CTIC:
Computational Topology in Image Context, LNCS, vol. 9667. 140–151.'
mla: Krcál, Marek, and Pawel Pilarczyk. *Computation of Cubical Steenrod Squares*.
Vol. 9667, Springer, 2016, pp. 140–51, doi:10.1007/978-3-319-39441-1_13.
short: M. Krcál, P. Pilarczyk, in:, Springer, 2016, pp. 140–151.
conference:
end_date: 2016-06-17
location: Marseille, France
name: 'CTIC: Computational Topology in Image Context'
start_date: 2016-06-15
date_created: 2018-12-11T11:50:52Z
date_published: 2016-06-02T00:00:00Z
date_updated: 2019-08-02T12:36:57Z
day: '02'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/978-3-319-39441-1_13
intvolume: ' 9667'
language:
- iso: eng
month: '06'
oa_version: None
page: 140 - 151
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication_status: published
publisher: Springer
publist_id: '6096'
quality_controlled: '1'
status: public
title: Computation of cubical Steenrod squares
type: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 9667
year: '2016'
...
---
_id: '1408'
abstract:
- lang: eng
text: 'The concept of well group in a special but important case captures homological
properties of the zero set of a continuous map (Formula presented.) on a compact
space K that are invariant with respect to perturbations of f. The perturbations
are arbitrary continuous maps within (Formula presented.) distance r from f for
a given (Formula presented.). The main drawback of the approach is that the computability
of well groups was shown only when (Formula presented.) or (Formula presented.).
Our contribution to the theory of well groups is twofold: on the one hand we improve
on the computability issue, but on the other hand we present a range of examples
where the well groups are incomplete invariants, that is, fail to capture certain
important robust properties of the zero set. For the first part, we identify a
computable subgroup of the well group that is obtained by cap product with the
pullback of the orientation of (Formula presented.) by f. In other words, well
groups can be algorithmically approximated from below. When f is smooth and (Formula
presented.), our approximation of the (Formula presented.)th well group is exact.
For the second part, we find examples of maps (Formula presented.) with all well
groups isomorphic but whose perturbations have different zero sets. We discuss
on a possible replacement of the well groups of vector valued maps by an invariant
of a better descriptive power and computability status.'
accept: '1'
acknowledgement: 'Open access funding provided by Institute of Science and Technology
(IST Austria). '
author:
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: Franek P, Krcál M. On computability and triviality of well groups. *Discrete
& Computational Geometry*. 2016;56(1):126-164. doi:10.1007/s00454-016-9794-2
apa: Franek, P., & Krcál, M. (2016). On computability and triviality of well
groups. *Discrete & Computational Geometry*, *56*(1), 126–164. https://doi.org/10.1007/s00454-016-9794-2
chicago: 'Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well
Groups.” *Discrete & Computational Geometry* 56, no. 1 (2016): 126–64.
https://doi.org/10.1007/s00454-016-9794-2.'
ieee: P. Franek and M. Krcál, “On computability and triviality of well groups,”
*Discrete & Computational Geometry*, vol. 56, no. 1, pp. 126–164, 2016.
ista: Franek P, Krcál M. 2016. On computability and triviality of well groups. Discrete
& Computational Geometry. 56(1), 126–164.
mla: Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups.”
*Discrete & Computational Geometry*, vol. 56, no. 1, Springer, 2016,
pp. 126–64, doi:10.1007/s00454-016-9794-2.
short: P. Franek, M. Krcál, Discrete & Computational Geometry 56 (2016) 126–164.
date_created: 2018-12-11T11:51:51Z
date_published: 2016-07-01T00:00:00Z
date_updated: 2019-11-14T08:41:55Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/s00454-016-9794-2
file:
- access_level: open_access
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:55Z
date_updated: 2018-12-12T10:10:55Z
file_id: '4846'
file_name: IST-2016-614-v1+1_s00454-016-9794-2.pdf
file_size: 905303
open_access: 1
relation: main_file
file_date_updated: 2018-12-12T10:10:55Z
intvolume: ' 56'
issue: '1'
language:
- iso: eng
month: '07'
oa_version: Published Version
page: 126 - 164
project:
- _id: 25F8B9BC-B435-11E9-9278-68D0E5697425
grant_number: M01980
name: Robust invariants of Nonlinear Systems
- _id: 25681D80-B435-11E9-9278-68D0E5697425
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: BFDF9788-01D1-11E9-AC17-EBD7A21D5664
name: IST Austria Open Access Fund
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5799'
pubrep_id: '614'
quality_controlled: '1'
related_material:
record:
- id: '1510'
relation: earlier_version
status: public
status: public
title: On computability and triviality of well groups
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 56
year: '2016'
...
---
_id: '1510'
abstract:
- lang: eng
text: 'The concept of well group in a special but important case captures homological
properties of the zero set of a continuous map f from K to R^n on a compact space
K that are invariant with respect to perturbations of f. The perturbations are
arbitrary continuous maps within L_infty distance r from f for a given r >
0. The main drawback of the approach is that the computability of well groups
was shown only when dim K = n or n = 1. Our contribution to the theory of well
groups is twofold: on the one hand we improve on the computability issue, but
on the other hand we present a range of examples where the well groups are incomplete
invariants, that is, fail to capture certain important robust properties of the
zero set. For the first part, we identify a computable subgroup of the well group
that is obtained by cap product with the pullback of the orientation of R^n by
f. In other words, well groups can be algorithmically approximated from below.
When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well
group is exact. For the second part, we find examples of maps f, f'' from K to
R^n with all well groups isomorphic but whose perturbations have different zero
sets. We discuss on a possible replacement of the well groups of vector valued
maps by an invariant of a better descriptive power and computability status. '
accept: '1'
alternative_title:
- LIPIcs
author:
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: 'Franek P, Krcál M. On computability and triviality of well groups. In: Vol
34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:842-856. doi:10.4230/LIPIcs.SOCG.2015.842'
apa: 'Franek, P., & Krcál, M. (2015). On computability and triviality of well
groups (Vol. 34, pp. 842–856). Presented at the SoCG: Symposium on Computational
Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SOCG.2015.842'
chicago: Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well
Groups,” 34:842–56. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.842.
ieee: 'P. Franek and M. Krcál, “On computability and triviality of well groups,”
presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands,
2015, vol. 34, pp. 842–856.'
ista: 'Franek P, Krcál M. 2015. On computability and triviality of well groups.
SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34. 842–856.'
mla: Franek, Peter, and Marek Krcál. *On Computability and Triviality of Well
Groups*. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015,
pp. 842–56, doi:10.4230/LIPIcs.SOCG.2015.842.
short: P. Franek, M. Krcál, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2015, pp. 842–856.
conference:
end_date: 2015-06-25
location: Eindhoven, Netherlands
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2015-06-22
date_created: 2018-12-11T11:52:26Z
date_published: 2015-06-11T00:00:00Z
date_updated: 2019-11-14T08:41:55Z
day: '11'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
doi: 10.4230/LIPIcs.SOCG.2015.842
file:
- access_level: open_access
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:13:19Z
date_updated: 2018-12-12T10:13:19Z
file_id: '5001'
file_name: IST-2016-503-v1+1_32.pdf
file_size: 623563
open_access: 1
relation: main_file
file_date_updated: 2018-12-12T10:13:19Z
intvolume: ' 34'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 842 - 856
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '5667'
pubrep_id: '503'
quality_controlled: '1'
related_material:
record:
- id: '1408'
relation: later_version
status: public
status: public
title: On computability and triviality of well groups
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2015'
...
---
_id: '1682'
abstract:
- lang: eng
text: 'We study the problem of robust satisfiability of systems of nonlinear equations,
namely, whether for a given continuous function f:K→ ℝn on a finite simplicial
complex K and α > 0, it holds that each function g: K → ℝn such that ||g -
f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps
into a sphere, we particularly show that this problem is decidable in polynomial
time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension
of previous computational applications of topological degree and related concepts
in numerical and interval analysis. Via a reverse reduction, we prove that the
problem is undecidable when dim K > 2n - 2, where the threshold comes from
the stable range in homotopy theory. For the lucidity of our exposition, we focus
on the setting when f is simplexwise linear. Such functions can approximate general
continuous functions, and thus we get approximation schemes and undecidability
of the robust satisfiability in other possible settings.'
article_number: '26'
author:
- first_name: Peter
full_name: Franek, Peter
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
citation:
ama: Franek P, Krcál M. Robust satisfiability of systems of equations. *Journal
of the ACM*. 2015;62(4). doi:10.1145/2751524
apa: Franek, P., & Krcál, M. (2015). Robust satisfiability of systems of equations.
*Journal of the ACM*, *62*(4). https://doi.org/10.1145/2751524
chicago: Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.”
*Journal of the ACM* 62, no. 4 (2015). https://doi.org/10.1145/2751524.
ieee: P. Franek and M. Krcál, “Robust satisfiability of systems of equations,” *Journal
of the ACM*, vol. 62, no. 4, 2015.
ista: Franek P, Krcál M. 2015. Robust satisfiability of systems of equations. Journal
of the ACM. 62(4).
mla: Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.”
*Journal of the ACM*, vol. 62, no. 4, 26, ACM, 2015, doi:10.1145/2751524.
short: P. Franek, M. Krcál, Journal of the ACM 62 (2015).
date_created: 2018-12-11T11:53:27Z
date_published: 2015-08-01T00:00:00Z
date_updated: 2019-01-24T13:04:05Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1145/2751524
intvolume: ' 62'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1402.0858
month: '08'
oa: 1
oa_version: Preprint
publication: Journal of the ACM
publication_status: published
publisher: ACM
publist_id: '5466'
quality_controlled: '1'
status: public
title: Robust satisfiability of systems of equations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 62
year: '2015'
...
---
_id: '2184'
abstract:
- lang: eng
text: 'Given topological spaces X,Y, a fundamental problem of algebraic topology
is understanding the structure of all continuous maps X→ Y. We consider a computational
version, where X,Y are given as finite simplicial complexes, and the goal is to
compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem
in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected;
in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical
tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and
simplicial sets) with algorithmic tools from effective algebraic topology (locally
effective simplicial sets and objects with effective homology). In contrast, [X,Y]
is known to be uncomputable for general X,Y, since for X = S1 it includes a well
known undecidable problem: testing triviality of the fundamental group of Y. In
follow-up papers, the algorithm is shown to run in polynomial time for d fixed,
and extended to other problems, such as the extension problem, where we are given
a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or
computing the Z2-index-everything in the stable range. Outside the stable range,
the extension problem is undecidable.'
acknowledgement: The research by M. K. was supported by project GAUK 49209. The research
by M. K. was also supported by project 1M0545 by the Ministry of Education of the
Czech Republic and by Center of Excellence { Inst. for Theor. Comput. Sci., Prague
(project P202/12/G061 of GACR). The research by U. W. was supported by the Swiss
National Science Foundation (SNF Projects 200021-125309, 200020-138230, and PP00P2-138948).
article_number: '17 '
author:
- first_name: Martin
full_name: Čadek, Martin
last_name: Čadek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Francis
full_name: Sergeraert, Francis
last_name: Sergeraert
- first_name: Lukáš
full_name: Vokřínek, Lukáš
last_name: Vokřínek
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing
all maps into a sphere. *Journal of the ACM*. 2014;61(3). doi:10.1145/2597629
apa: Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner,
U. (2014). Computing all maps into a sphere. *Journal of the ACM*, *61*(3).
https://doi.org/10.1145/2597629
chicago: Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek,
and Uli Wagner. “Computing All Maps into a Sphere.” *Journal of the ACM*
61, no. 3 (2014). https://doi.org/10.1145/2597629.
ieee: M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner,
“Computing all maps into a sphere,” *Journal of the ACM*, vol. 61, no. 3,
2014.
ista: Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2014. Computing
all maps into a sphere. Journal of the ACM. 61(3).
mla: Čadek, Martin, et al. “Computing All Maps into a Sphere.” *Journal of the
ACM*, vol. 61, no. 3, 17, ACM, 2014, doi:10.1145/2597629.
short: M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, Journal
of the ACM 61 (2014).
date_created: 2018-12-11T11:56:12Z
date_published: 2014-05-01T00:00:00Z
date_updated: 2019-01-24T13:07:01Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1145/2597629
intvolume: ' 61'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1105.6257
month: '05'
oa: 1
oa_version: Preprint
publication: Journal of the ACM
publication_status: published
publisher: ACM
publist_id: '4797'
quality_controlled: '1'
status: public
title: Computing all maps into a sphere
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 61
year: '2014'
...
---
_id: '1842'
abstract:
- lang: eng
text: We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2
outerplanar triangulations in both convex and general cases. We also prove that
the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by
O(n3) and O(n10), in the convex and general case, respectively. We then apply
similar methods to prove an (Formula presented.) upper bound on the Ramsey number
of a path with n ordered vertices.
acknowledgement: Marek Krčál was supported by the ERC Advanced Grant No. 267165.
author:
- first_name: Josef
full_name: Cibulka, Josef
last_name: Cibulka
- first_name: Pu
full_name: Gao, Pu
last_name: Gao
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Tomáš
full_name: Valla, Tomáš
last_name: Valla
- first_name: Pavel
full_name: Valtr, Pavel
last_name: Valtr
citation:
ama: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. On the geometric ramsey number
of outerplanar graphs. *Discrete & Computational Geometry*. 2014;53(1):64-79.
doi:10.1007/s00454-014-9646-x
apa: Cibulka, J., Gao, P., Krcál, M., Valla, T., & Valtr, P. (2014). On the
geometric ramsey number of outerplanar graphs. *Discrete & Computational
Geometry*, *53*(1), 64–79. https://doi.org/10.1007/s00454-014-9646-x
chicago: 'Cibulka, Josef, Pu Gao, Marek Krcál, Tomáš Valla, and Pavel Valtr. “On
the Geometric Ramsey Number of Outerplanar Graphs.” *Discrete & Computational
Geometry* 53, no. 1 (2014): 64–79. https://doi.org/10.1007/s00454-014-9646-x.'
ieee: J. Cibulka, P. Gao, M. Krcál, T. Valla, and P. Valtr, “On the geometric ramsey
number of outerplanar graphs,” *Discrete & Computational Geometry*, vol.
53, no. 1, pp. 64–79, 2014.
ista: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. 2014. On the geometric ramsey
number of outerplanar graphs. Discrete & Computational Geometry. 53(1), 64–79.
mla: Cibulka, Josef, et al. “On the Geometric Ramsey Number of Outerplanar Graphs.”
*Discrete & Computational Geometry*, vol. 53, no. 1, Springer, 2014,
pp. 64–79, doi:10.1007/s00454-014-9646-x.
short: J. Cibulka, P. Gao, M. Krcál, T. Valla, P. Valtr, Discrete & Computational
Geometry 53 (2014) 64–79.
date_created: 2018-12-11T11:54:18Z
date_published: 2014-11-14T00:00:00Z
date_updated: 2019-08-02T12:37:28Z
day: '14'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/s00454-014-9646-x
intvolume: ' 53'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1310.7004
month: '11'
oa: 1
oa_version: Submitted Version
page: 64 - 79
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5260'
status: public
title: On the geometric ramsey number of outerplanar graphs
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 53
year: '2014'
...
---
_id: '2807'
abstract:
- lang: eng
text: 'We consider several basic problems of algebraic topology, with connections
to combinatorial and geometric questions, from the point of view of computational
complexity. The extension problem asks, given topological spaces X; Y , a subspace
A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X
→ Y . For computational purposes, we assume that X and Y are represented as finite
simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map.
In this generality the problem is undecidable, as follows from Novikov''s result
from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study
the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected;
informally, this means that Y has \no holes up to dimension k-1" (a basic
example of such a Y is the sphere Sk). We prove that, on the one hand, this problem
is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2,
we obtain an algorithm that solves the extension problem in polynomial time assuming
Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides
a classification of all extensions up to homotopy (continuous deformation). This
relies on results of our SODA 2012 paper, and the main new ingredient is a machinery
of objects with polynomial-time homology, which is a polynomial-time analog of
objects with effective homology developed earlier by Sergeraert et al. We also
consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected
Y . Their computability was established by Brown in 1957; we show that πk(Y )
can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick
proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where
Y is a cell complex with certain rather compact encoding. We strengthen his result
to #P-hardness for Y given as a simplicial complex. '
accept: '1'
author:
- first_name: Martin
full_name: Čadek, Martin
last_name: Čadek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Lukáš
full_name: Vokřínek, Lukáš
last_name: Vokřínek
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Čadek M, Krcál M, Matoušek J, Vokřínek L, Wagner U. Extending continuous maps:
Polynomiality and undecidability. In: *45th Annual ACM Symposium on Theory of
Computing*. ACM; 2013:595-604. doi:10.1145/2488608.2488683'
apa: 'Čadek, M., Krcál, M., Matoušek, J., Vokřínek, L., & Wagner, U. (2013).
Extending continuous maps: Polynomiality and undecidability. In *45th Annual
ACM Symposium on theory of computing* (pp. 595–604). Palo Alto, CA, United
States: ACM. https://doi.org/10.1145/2488608.2488683'
chicago: 'Čadek, Martin, Marek Krcál, Jiří Matoušek, Lukáš Vokřínek, and Uli Wagner.
“Extending Continuous Maps: Polynomiality and Undecidability.” In *45th Annual
ACM Symposium on Theory of Computing*, 595–604. ACM, 2013. https://doi.org/10.1145/2488608.2488683.'
ieee: 'M. Čadek, M. Krcál, J. Matoušek, L. Vokřínek, and U. Wagner, “Extending continuous
maps: Polynomiality and undecidability,” in *45th Annual ACM Symposium on theory
of computing*, Palo Alto, CA, United States, 2013, pp. 595–604.'
ista: 'Čadek M, Krcál M, Matoušek J, Vokřínek L, Wagner U. 2013. Extending continuous
maps: Polynomiality and undecidability. 45th Annual ACM Symposium on theory of
computing. STOC: Symposium on the Theory of Computing 595–604.'
mla: 'Čadek, Martin, et al. “Extending Continuous Maps: Polynomiality and Undecidability.”
*45th Annual ACM Symposium on Theory of Computing*, ACM, 2013, pp. 595–604,
doi:10.1145/2488608.2488683.'
short: M. Čadek, M. Krcál, J. Matoušek, L. Vokřínek, U. Wagner, in:, 45th Annual
ACM Symposium on Theory of Computing, ACM, 2013, pp. 595–604.
conference:
end_date: 2013-06-04
location: Palo Alto, CA, United States
name: 'STOC: Symposium on the Theory of Computing'
start_date: 2013-06-01
date_created: 2018-12-11T11:59:42Z
date_published: 2013-06-01T00:00:00Z
date_updated: 2019-08-02T12:37:51Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1145/2488608.2488683
file:
- access_level: open_access
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:14:29Z
date_updated: 2018-12-12T10:14:29Z
file_id: '5081'
file_name: IST-2016-533-v1+1_Extending_continuous_maps_polynomiality_and_undecidability.pdf
file_size: 447945
open_access: 1
relation: main_file
file_date_updated: 2018-12-12T10:14:29Z
language:
- iso: eng
month: '06'
oa: 1
oa_version: Submitted Version
page: 595 - 604
publication: 45th Annual ACM Symposium on theory of computing
publication_status: published
publisher: ACM
publist_id: '4078'
pubrep_id: '533'
quality_controlled: '1'
status: public
title: 'Extending continuous maps: Polynomiality and undecidability'
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2013'
...
---
_id: '2440'
abstract:
- lang: eng
text: We present an algorithm for computing [X, Y], i.e., all homotopy classes of
continuous maps X → Y, where X, Y are topological spaces given as finite simplicial
complexes, Y is (d - 1)-connected for some d ≥ 2 (for example, Y can be the d-dimensional
sphere S d), and dim X ≤ 2d - 2. These conditions on X, Y guarantee that [X, Y]
has a natural structure of a finitely generated Abelian group, and the algorithm
finds generators and relations for it. We combine several tools and ideas from
homotopy theory (such as Postnikov systems, simplicial sets, and obstruction theory)
with algorithmic tools from effective algebraic topology (objects with effective
homology). We hope that a further extension of the methods developed here will
yield an algorithm for computing, in some cases of interest, the ℤ 2-index, which
is a quantity playing a prominent role in Borsuk-Ulam style applications of topology
in combinatorics and geometry, e.g., in topological lower bounds for the chromatic
number of a graph. In a certain range of dimensions, deciding the embeddability
of a simplicial complex into ℝ d also amounts to a ℤ 2-index computation. This
is the main motivation of our work. We believe that investigating the computational
complexity of questions in homotopy theory and similar areas presents a fascinating
research area, and we hope that our work may help bridge the cultural gap between
algebraic topology and theoretical computer science.
author:
- first_name: Martin
full_name: Čadek, Martin
last_name: Čadek
- first_name: Marek
full_name: Marek Krcál
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Francis
full_name: Sergeraert, Francis
last_name: Sergeraert
- first_name: Lukáš
full_name: Vokřínek, Lukáš
last_name: Vokřínek
- first_name: Uli
full_name: Uli Wagner
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing
all maps into a sphere. In: SIAM; 2012:1-10.'
apa: 'Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner,
U. (2012). Computing all maps into a sphere (pp. 1–10). Presented at the SODA:
Symposium on Discrete Algorithms, SIAM.'
chicago: Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek,
and Uli Wagner. “Computing All Maps into a Sphere,” 1–10. SIAM, 2012.
ieee: 'M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner,
“Computing all maps into a sphere,” presented at the SODA: Symposium on Discrete
Algorithms, 2012, pp. 1–10.'
ista: 'Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2012. Computing
all maps into a sphere. SODA: Symposium on Discrete Algorithms 1–10.'
mla: Čadek, Martin, et al. *Computing All Maps into a Sphere*. SIAM, 2012,
pp. 1–10.
short: M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, in:,
SIAM, 2012, pp. 1–10.
conference:
name: 'SODA: Symposium on Discrete Algorithms'
date_created: 2018-12-11T11:57:40Z
date_published: 2012-01-01T00:00:00Z
date_updated: 2019-04-26T07:22:13Z
day: '01'
extern: 1
main_file_link:
- open_access: '0'
url: http://arxiv.org/abs/1105.6257
month: '01'
page: 1 - 10
publication_status: published
publisher: SIAM
publist_id: '4466'
quality_controlled: 0
status: public
title: Computing all maps into a sphere
type: conference
year: '2012'
...