Abstract

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.

MSC codes

  1. 90C27
  2. 06A07
  3. 52B99
  4. 57Q05
  5. 57R70

Keywords

  1. discrete Morse function
  2. Morse matching

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References

1.
T. Achterberg, SCIP—A framework to integrate constraint and mixed integer programming, ZIB‐Report 04‐19, Zuse Institute Berlin, Berlin, Germany, 2004.
2.
Tobias Achterberg, Thorsten Koch, Alexander Martin, Branching rules revisited, Oper. Res. Lett., 33 (2005), 42–54
3.
E. Babson, A. Björner, S. Linusson, J. Shareshian, and V. Welker, Complexes of not i‐connected graphs, Topology, 38 (1999), pp. 271–299.
4.
E. Batzies, V. Welker, Discrete Morse theory for cellular resolutions, J. Reine Angew. Math., 543 (2002), 147–168
5.
Manoj Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math., 217 (2000), 101–113, Formal power series and algebraic combinatorics (Vienna, 1997)
6.
Manoj Chari, Michael Joswig, Complexes of discrete Morse functions, Discrete Math., 302 (2005), 39–51
7.
Ömer Eğecioğlu, Teofilo Gonzalez, A computationally intractable problem on simplicial complexes, Comput. Geom., 6 (1996), 85–98
8.
Robin Forman, Morse theory for cell complexes, Adv. Math., 134 (1998), 90–145
9.
Robin Forman, Morse theory and evasiveness, Combinatorica, 20 (2000), 489–504
10.
Robin Forman, A user’s guide to discrete Morse theory, Sém. Lothar. Combin., 48 (2002), 0–0Art. B48c, 35 pp. (electronic)
11.
Ewgenij Gawrilow, Michael Joswig, polymake: a framework for analyzing convex polytopes, DMV Sem., Vol. 29, Birkhäuser, Basel, 2000, 43–73
12.
E. Gawrilow and M. Joswig, polymake: Version 2.1.0, http://www.math.tu‐berlin.de/polymake, 2004. With contributions by T. Schröder and N. Witte.
13.
Martin Grötschel, Michael Jünger, Gerhard Reinelt, A cutting plane algorithm for the linear ordering problem, Oper. Res., 32 (1984), 1195–1220
14.
Martin Grötschel, Michael Jünger, Gerhard Reinelt, On the acyclic subgraph polytope, Math. Programming, 33 (1985), 28–42
15.
M. Hachimori, Simplicial complex library. Available online from http://infoshako.sk.tsukuba.ac.jp/˜hachi/math/library/index_eng.html, 2001.
16.
P. Hammer, E. Johnson, U. Peled, Facets of regular 0‐1 polytopes, Math. Programming, 8 (1975), 179–206
17.
Patricia Hersh, On optimizing discrete Morse functions, Adv. in Appl. Math., 35 (2005), 294–322
18.
Costas Iliopoulos, Worst‐case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix, SIAM J. Comput., 18 (1989), 658–669
19.
Jakob Jonsson, On the topology of simplicial complexes related to 3‐connected and Hamiltonian graphs, J. Combin. Theory Ser. A, 104 (2003), 169–199
20.
M. Joswig, Computing invariants of simplicial manifolds, preprint. Available online from math.AT/0401176, 2004.
21.
M. Jünger, Polyhedral combinatorics and the acyclic subdigraph problem, Research and Exposition in Mathematics, Vol. 7, Heldermann Verlag, 1985x+128
22.
Bernhard Korte, Jens Vygen, Combinatorial optimization, Algorithms and Combinatorics, Vol. 21, Springer‐Verlag, 2002xiv+530, Theory and algorithms
23.
Thomas Lewiner, Hélio Lopes, Geovan Tavares, Optimal discrete Morse functions for 2‐manifolds, Comput. Geom., 26 (2003), 221–233
24.
Thomas Lewiner, Hélio Lopes, Geovan Tavares, Toward optimality in discrete Morse theory, Experiment. Math., 12 (2003), 271–285
25.
Frank Lutz, Small examples of nonconstructible simplicial balls and spheres, SIAM J. Discrete Math., 18 (2004), 103–109
26.
F. H. Lutz, A vertex‐minimal nonshellable simplicial 3‐ball with 9 vertices and 18 facets, Electronic Geometry Models, (2004). Available online from www.eg‐models.de.
27.
James Munkres, Elements of algebraic topology, Addison‐Wesley Publishing Company, 1984ix+454
28.
G. Reinelt, The linear ordering problem: algorithms and applications, Research and Exposition in Mathematics, Vol. 8, Heldermann Verlag, 1985xi+158
29.
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, 24, Springer‐Verlag, Berlin, 2003.
30.
John Shareshian, Discrete Morse theory for complexes of 2‐connected graphs, Topology, 40 (2001), 681–701
31.
D. M. Warme, P. Winter, and M. Zachariasen, Exact solutions to large‐scale plane steiner tree problems, in Proceedings of the 10th Annual ACM‐SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1999, pp. 979–980.

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 11 - 25
ISSN (online): 1095-7146

History

Published online: 1 August 2006

MSC codes

  1. 90C27
  2. 06A07
  3. 52B99
  4. 57Q05
  5. 57R70

Keywords

  1. discrete Morse function
  2. Morse matching

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