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## COMPUTING OPTIMAL MORSE MATCHINGS (2004)

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### Citations

903 | Combinatorial Optimization: Polyhedra and Efficiency, - Schrijver - 2003 |

347 | Computing discrete minimal surfaces and their conjugates,
- Pinkall, Juni, et al.
- 1993
(Show Context)
Citation Context ...inatorial cores of known phenomena in differential geometry are investigated, while keeping a steady eye on applications in computer graphics and mathematical physics; see, e.g., Pinkall and Polthier =-=[28]-=-, Bobenko and Suris [5]. We firmly believe that —via discrete Morse theory— techniques from combinatorial optimization are highly relevant to these topics. Our contribution is a first step. The paper ... |

308 |
Combinatorial Optimization: Theory and algorithms,
- Korte, Vygen
- 2008
(Show Context)
Citation Context ...iolates the cycle inequality (6) if and only if ˜ ℓ(C ′ ) < 1. Since ˜ ℓ(e ′ ) ≥ 0, we can use the Floyd-Warshall algorithm to solve the separation problem in time O � |V ′ | 3� ; see Korte and Vygen =-=[23]-=-. If G = Hi and W is the part arising from the higher dimensional faces, we have |V ′ | = �i+2� 2 |W | = �i+2� 2 fi+1. This leads to an O � (d + 1) 6n3� algorithm for separating cycle inequalities, wh... |

233 | A discrete Morse theory for cell complexes.
- Forman
- 1995
(Show Context)
Citation Context ...mming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results. 1. Introduction Discrete Morse theory was developed by Forman =-=[9, 11]-=- as a combinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and ... |

209 | Kneser’s conjecture, chromatic number, and homotopy. - Lovász - 1978 |

203 |
Munkres. Elements of Algebraic Topology
- R
(Show Context)
Citation Context ... complex and M be a Morse matching on ∆. Then ∆ is homotopy equivalent to a CW-complex containing a cell of dimension i for each critical face of dimension i.s4 JOSWIG AND PFETSCH We refer to Munkres =-=[27]-=- for more information on CW-complexes. By this result we can hope for a compact representation of the topology of ∆ (up to homotopy) by computing a Morse matching with few critical faces. This is the ... |

168 | polymake: A framework for analyzing convex polytopes. In:
- Gawrilow, Joswig
- 2000
(Show Context)
Citation Context ...nch-and-cut algorithm along the lines of Section 5. The C++ implementation uses the framework SCIP (Solving Constraint Integer Programs) by Achterberg, see [1]. It furthermore builds on polymake; see =-=[12, 13]-=-. As an LP solver we used CPLEX 9.0. As the basis of our implementation we take the formulation of MaxMM in Section 5. Matching inequalities (5) and Betti inequalities (8) (together with variable boun... |

100 |
A Cutting Plane Algorithm for the Linear Ordering Problem.
- Grotschel, Junger, et al.
- 1984
(Show Context)
Citation Context ...orresponding problem for ASP; see Section 5.2. Furthermore, there is a similarity to the relation between the ASP and the linear ordering problem (see Reinelt [29], and Grötschel, Jünger, and Reinelt =-=[14]-=-): an alternative formulation for our problem can be obtained by adding matching inequalities to the linear ordering formulation; this directly models discrete Morse functions as linear orderings of t... |

82 | Integrable systems on quad-graphs,
- Bobenko, Suris
- 2002
(Show Context)
Citation Context ... phenomena in differential geometry are investigated, while keeping a steady eye on applications in computer graphics and mathematical physics; see, e.g., Pinkall and Polthier [28], Bobenko and Suris =-=[5]-=-. We firmly believe that —via discrete Morse theory— techniques from combinatorial optimization are highly relevant to these topics. Our contribution is a first step. The paper is structured as follow... |

67 |
On discrete Morse functions and combinatorial decompositions, Discrete Math
- Chari
(Show Context)
Citation Context ...ck to the simplicial setting, however, to simplify the presentation. 2. Discrete Morse Functions and Morse Matchings We will first introduce discrete Morse functions as developed by Forman [9]. Chari =-=[6]-=- showed that the essential structure of discrete Morse functions is captured by so-called Morse matchings. It turns out that this latter formulation directly leads to a combinatorial optimization prob... |

52 |
The Linear Ordering Problem: Algorithms and Applications.
- Reinelt
- 1985
(Show Context)
Citation Context ..., however, is more complicated than the corresponding problem for ASP; see Section 5.2. Furthermore, there is a similarity to the relation between the ASP and the linear ordering problem (see Reinelt =-=[29]-=-, and Grötschel, Jünger, and Reinelt [14]): an alternative formulation for our problem can be obtained by adding matching inequalities to the linear ordering formulation; this directly models discrete... |

48 | 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.
- Iliopoulos
- 1989
(Show Context)
Citation Context ...ers overÉand finite fields can easily be obtained in polynomial time (in the size of ∆), by computing the ranks of the boundary matrices for each dimension. Although harder to compute (see Iliopoulos =-=[19]-=-), the homology over�can be used to choose among the finite fields orÉ, in order to obtain the strongest form of the Morse inequalities (3.3). 4. Hardness of Optimal Morse Matchings In this section we... |

37 |
Facets of regular 0-1 polytopes.
- Hammer, Johnson, et al.
- 1975
(Show Context)
Citation Context ...se matching. It is well known that this implies that every facet defining inequality α T x ≤ β not equivalent to the non-negativity inequalities fulfills: α ≥ 0, β > 0; see Hammer, Johnson, and Peled =-=[17]-=-. Interestingly, if we generalize Morse matchings to acyclic matchings for arbitrary graphs, the collection of such acyclic matchings is not necessarily monotone anymore; see the example in Figure 2. ... |

34 | Discrete Morse theory for cellular resolutions,
- Batzies, Welker
- 2002
(Show Context)
Citation Context ...ombinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and Welker =-=[4]-=-, and Jonsson [20]. It turns out that the topologically relevant information of a discrete Morse function f on a simplicial complex can be encoded as a (partial) matching in its Hasse diagram (conside... |

33 | SCIP - a framework to integrate Constraint and Mixed Integer Programming
- Achterberg
- 2004
(Show Context)
Citation Context ...ourse, there are exponentially many cycle inequalities (6). Hence we have to deal with the separation problem for these inequalities. For the separation problem, we can assume that we are given x ∗ ∈ =-=[0,1]-=- A , which satisfies all matching inequalities (5). We consider the separation for each graph Hi in turn, i = 0,... ,d − 1. The problem is to find an undirected cycle C in Hi such that x ∗ (C) > 1 2 |... |

25 |
On the acyclic subgraph polytope.
- Grotschel, Junger, et al.
- 1985
(Show Context)
Citation Context ...n consists of two parts: one for the matching conditions and one for the acyclicity constraints. This turns out to be related to the acyclic subgraph problem studied by Grötschel, Jünger, and Reinelt =-=[15]-=-. We derive polyhedral results for the corresponding polytope. In particular, we give two different polynomial time algorithms for the separation of the acyclicity constraints. The paper closes with c... |

21 | The complexes of not i-connected graphs
- Babson, Björner, et al.
- 1999
(Show Context)
Citation Context ...as developed by Forman [9, 11] as a combinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. =-=[3]-=-, Forman [10], Batzies and Welker [4], and Jonsson [20]. It turns out that the topologically relevant information of a discrete Morse function f on a simplicial complex can be encoded as a (partial) m... |

14 | Small examples of nonconstructible simplicial balls and spheres
- Lutz
(Show Context)
Citation Context ...e considered the following complexes: CP2 (complex projective plane), CP2+CP2 (connected sum of CP2 with itself), MNSB and MNSS ((vertex) minimal non-shellable ball and sphere, respectively; see Lutz =-=[26]-=-). All computational experiments were run on a 3 GHz Pentium machine running Linux. In the tables of computational results, n denotes the number of faces, m the number of arcs in the HassesCOMPUTING O... |

12 |
Polyhedral combinatorics and the acyclic subdigraph problem, Heldermann
- Jünger
- 1985
(Show Context)
Citation Context ...ield in Theorem 3.3 to employ the Morse inequalities in their strongest form. Note. The cycle inequalities (6) are similar to the cycle inequalities for the acyclic subgraph problem (ASP); see Jünger =-=[22]-=-, and Grötschel, Jünger, and Reinelt [15]. The separation problem for (6), however, is more complicated than the corresponding problem for ASP; see Section 5.2. Furthermore, there is a similarity to t... |

11 | On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs
- Jonsson
(Show Context)
Citation Context ...og to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and Welker [4], and Jonsson =-=[20]-=-. It turns out that the topologically relevant information of a discrete Morse function f on a simplicial complex can be encoded as a (partial) matching in its Hasse diagram (considered as a graph), t... |

10 |
A computationally intractable problem on simplicial complexes
- Eǧecioǧlu, Gonzalez
- 1996
(Show Context)
Citation Context ... maximal cardinality. As we saw previously, this is equivalent to minimize the number of critical faces. We want to reduce the following collapsibility problem, introduced by Eˇgecioˇglu and Gonzalez =-=[8]-=-, to the problem of finding an optimal Morse matching: Given a connected pure 2-dimensional simplicial complex ∆, which is embeddable inÊ3 and an integer k, decide whether there exists a subset K of t... |

9 | On optimizing discrete Morse functions,
- Hersh
- 2005
(Show Context)
Citation Context ... undecidable even for highly structured classes of topological spaces, such as smooth 4-manifolds. Optimization of discrete Morse matchings has been studied by Lewiner, Lopes, and Tavares [24]. Hersh =-=[18]-=- investigated heuristic approaches to the maximum Morse matching problem with applications to combinatorics. Morse matchings can also be interpreted as pivoting strategies for homology computations; s... |

8 |
Complexes of discrete Morse functions,”
- Chari, Joswig
- 2005
(Show Context)
Citation Context ...interpreted as pivoting strategies for homology computations; see [21]. Furthermore, the set of all Morse matchings of a given simplicial complex itself has the structure of a simplicial complex; see =-=[7]-=-. Since its beginnings in Lovász’ proof [25] of the Kneser Conjecture, combinatorial topology seeks to solve combinatorial problems with techniques from (primarily algebraic) topology. And, conversely... |

8 | Morse theory and evasiveness - Forman |

8 | Computing invariants of simplicial manifolds
- Joswig
- 2004
(Show Context)
Citation Context ...vestigated heuristic approaches to the maximum Morse matching problem with applications to combinatorics. Morse matchings can also be interpreted as pivoting strategies for homology computations; see =-=[21]-=-. Furthermore, the set of all Morse matchings of a given simplicial complex itself has the structure of a simplicial complex; see [7]. Since its beginnings in Lovász’ proof [25] of the Kneser Conjectu... |

6 |
Simplicial complex Library, http://infoshako.sk.tsukuba.ac.jp/∼hachi/math/library/index eng.html
- Hachimori
(Show Context)
Citation Context ...acyclic matching, because the path is unique. This heuristic turns out to be extremely successful; see below. We tested the implementation on a set of simplicial complexes collected by Hachimori; see =-=[16]-=- for more details. Additionally, we considered the following complexes: CP2 (complex projective plane), CP2+CP2 (connected sum of CP2 with itself), MNSB and MNSS ((vertex) minimal non-shellable ball a... |

5 |
Toward Optimality in Discrete Morse Theory.” Experimental Math. 12 (2003), 271–285. Henry King
- Lewiner, Lopes, et al.
- 2005
(Show Context)
Citation Context ... known to be undecidable even for highly structured classes of topological spaces, such as smooth 4-manifolds. Optimization of discrete Morse matchings has been studied by Lewiner, Lopes, and Tavares =-=[24]-=-. Hersh [18] investigated heuristic approaches to the maximum Morse matching problem with applications to combinatorics. Morse matchings can also be interpreted as pivoting strategies for homology com... |

1 |
Branching rules revisited. ZIB-Report
- Achterberg, Koch, et al.
- 2004
(Show Context)
Citation Context ...ribed in Section 5.2.2. Additionally, Gomory cuts are added. As a branching rule we use reliability branching implemented in SCIP, a variable branching rule introduced by Achterberg, Koch, and Martin =-=[2]-=-. We implemented the following primal heuristic. First a simple greedy algorithm is run: We start with the empty matching M = ∅. We add arcs of the Hasse diagram to M in the order of decreasing value ... |

1 |
Simplicial complex library.http://infoshako.sk.tsukuba.ac.jp/~hachi/math/library/index_eng.html
- Hachimori
- 2001
(Show Context)
Citation Context ...acyclic matching, because the path is unique. This heuristic turns out to be extremely successful; see below. We tested the implementation on a set of simplicial complexes collected by Hachimori; see =-=[16]-=- for more details. Additionally, we considered the following complexes: CP2 (complex projective plane), CP2+CP2 (connected sum of CP2 with itself), MNSB and MNSS ((vertex) minimal non-shellable ball a... |