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COMPUTING OPTIMAL MORSE MATCHINGS
, 2004
"... Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computation ..."
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Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.
Computational topology
 Algorithms and Theory of Computation Handbook
, 2010
"... According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in t ..."
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According to the Oxford English Dictionary, the word topology is derived of topos ( � ) meaning place, andlogy ( ���), a variant of the verb ´��� � , meaning to speak. As such, topology speaks about places: how local neighborhoods connect to each other to form a space. Computational topology, in turn, undertakes the challenge of studying topology using a computer.
Computing Optimal Discrete Morse Functions (Extended Abstract)
, 2004
"... The essential structural information of discrete Morse functions is captured by socalled Morse matchings. We show that computing optimal Morse matchings is N Phard and give an integer programming formulation for the problem. Then we present first polyhedral results for the corresponding polytope a ..."
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The essential structural information of discrete Morse functions is captured by socalled Morse matchings. We show that computing optimal Morse matchings is N Phard and give an integer programming formulation for the problem. Then we present first polyhedral results for the corresponding polytope and report on some preliminary computational results.
Hamiltonian submanifolds of regular polytopes
 Discr. & Comput. Geom
"... Abstract: We investigate polyhedral 2kmanifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex kHamiltonian if it contains the full kskeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously stu ..."
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Abstract: We investigate polyhedral 2kmanifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex kHamiltonian if it contains the full kskeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are socalled superneighborly triangulations) we focus on the case of the cross polytope and the sporadic regular 4polytopes. By our results the existence of 1Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2Hamiltonian 4manifolds in the ddimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of S 2 × S 2. By this example all regular cases of n vertices with n < 20 or, equivalently, all cases of regular dpolytopes with d ≤ 9 are now decided.
Geometric Reasoning with polymake
, 2005
"... The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of ..."
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The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background.