## Some Algorithmic Problems in Polytope Theory (2003)

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Venue: | IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS |

Citations: | 11 - 1 self |

### BibTeX

@INPROCEEDINGS{Kaibel03somealgorithmic,

author = {Volker Kaibel and Marc E. Pfetsch},

title = {Some Algorithmic Problems in Polytope Theory},

booktitle = {IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS},

year = {2003},

pages = {23--47},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

### Citations

10959 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...e use without explanation we refer to Ziegler's book [65]. Similarly, for the concepts from the theory of computational complexity that play a role here we refer to Garey and Johnson's classical text =-=[24]-=-. Whenever we talk about polynomial reductions this refers to polynomial time Turing-reductions. For some of the problems Some Algorithmic Problems in Polytope Theory 3 the output can be exponentially... |

652 | A new polynomial-time algorithm for linear programming
- Karmarkar
- 1984
(Show Context)
Citation Context ...he first polynomial time algorithm was a variant of the ellipsoid algorithm due to Khachiyan [38]. Later, also interior point methods solving the problem in polynomial time were discovered (Karmarkar =-=[37]-=-). Megiddo found an algorithm solving the problem for a fixed number d of variables in O(m) arithmetic operations (Megiddo [44]). No strongly polynomial time algorithm (performing a number of arithmet... |

421 |
The complexity of enumeration and reliability problems
- Valiant
- 1979
(Show Context)
Citation Context ...sider an instance of SAT, i.e., a formula in conjunctive normal form (CNFformula) C1∧· · ·∧Cm with variables x1, . . . , xn (each Ci contains only disjunctions of literals). It is well known (Valiant =-=[63]-=-) that computing the number of satisfying truth assignments is #P-complete. Define E = {t1, f1, . . .,tn, fn}. Part I. First, let E be the vertex set of a simplicial complex ∆ defined by the minimal n... |

378 |
A polynomial algorithm in linear programming
- Khachiyan
- 1979
(Show Context)
Citation Context ...rongly polynomial time algorithm known Status (fixed dim.): Linear time in m (the number of inequalities) The first polynomial time algorithm was a variant of the ellipsoid algorithm due to Khachiyan =-=[38]-=-. Later, also interior point methods solving the problem in polynomial time were discovered (Karmarkar [37]). Megiddo found an algorithm solving the problem for a fixed number d of variables in O(m) a... |

361 |
Lectures on Polytopes
- Ziegler
- 1995
(Show Context)
Citation Context ...idered to be “folklore” or “easy to prove.” At the end related problems in this paper are listed. For all notions in the theory of polytopes that we use without explanation we refer to Ziegler’s book =-=[65]-=-. Similarly, for the concepts from the theory of computational complexity that play a role here we refer to Garey and Johnson’s classical text [24]. Whenever we talk about polynomial reductions this r... |

256 |
On the foundations of combinatorial theory I. Theory of Möbius functions
- Rota
- 1964
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Citation Context ...ntly the fastest way to compute the Euler characteristic is to determine V = {S : S is an intersection of facets of ∆} and then compute χ(∆) in time O � |V| 2� by a Möbius function approach, see Rota =-=[54]-=-. Usually V is much smaller than the whole face lattice of ∆. V can be listed lexicographically by an algorithm of Ganter [23], in time O(min{m, n} · α · |V|), where α is the number of vertex-facets i... |

194 | Linear programming in linear time when the dimension is fixed
- Megiddo
- 1984
(Show Context)
Citation Context ...ethods solving the problem in polynomial time were discovered (Karmarkar [37]). Megiddo found an algorithm solving the problem for a fixed number d of variables in O(m) arithmetic operations (Megiddo =-=[44]-=-). No strongly polynomial time algorithm (performing a number of arithmetic operations that is bounded polynomially in d and the number of half-spaces rather than in the coding lengths of the input co... |

182 | A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
- Avis, Fukuda
- 1992
(Show Context)
Citation Context ...rithms which are faster than Chazelle’s algorithm for small n, e.g., an O � m log n + (mn) 1−1/(⌊d/2⌋+1) polylog m � algorithm of Chan [9]. For general d, the reverse search method of Avis and Fukuda =-=[2]-=- solves the problem for simple polyhedra in polynomial total time, using working spacesSome Algorithmic Problems in Polytope Theory 5 (without space for output) bounded polynomially in the input size.... |

171 |
Dividing a Graph into Triconnected Components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ...graph G, test if the facetss18 Volker Kaibel and Marc E. Pfetsch can consistently be embedded in the plane (linear time [30,45]) and check for 3-connectedness (in linear time, see Hopcroft and Tarjan =-=[29]-=-). Mnëv proved that the Steinitz Problem for d-polytopes with d +4 vertices is N P-hard [47]. Even more, Richter-Gebert [53] proved that for (fixed) d ≥ 4 the problem is N P-hard. For fixed d ≥ 4 it i... |

166 | A subexponential bound for linear programming, Algorithmica 16 (4–5
- Matoušek, Sharir, et al.
- 1996
(Show Context)
Citation Context ...l time variant of the simplex algorithm is known. However, a randomized version of the simplex algorithm solves the problem in (expected) subexponential time (Kalai [36], Matouˇsek, Sharir, and Welzl =-=[42]-=-). Related problems: 25, 26, 27s16 Volker Kaibel and Marc E. Pfetsch 25. Optimal Vertex Input: H-description of a polyhedron P ⊂Éd , c ∈Éd Output: inf � c T v | v vertex of P � ∈É∪{∞} and, if the infi... |

147 |
The maximum numbers of faces of a convex polytope, Mathematika 17
- McMullen
- 1970
(Show Context)
Citation Context ... Enumeration is also strongly polynomially equivalent to Problem 2. For fixed d, Chazelle [12] found an O � m ⌊d/2⌋� polynomial time algorithm, which is optimal by the Upper Bound Theorem of McMullen =-=[43]-=-. There exist algorithms which are faster than Chazelle’s algorithm for small n, e.g., an O � m log n + (mn) 1−1/(⌊d/2⌋+1) polylog m � algorithm of Chan [9]. For general d, the reverse search method o... |

141 |
Isomorphism of graphs of bounded valence can be tested in polynomial time
- Luks
- 1982
(Show Context)
Citation Context ...constant dimension the problem can be solved in polynomial time by a reduction [34] to the graph isomorphism problem for graphs of bounded degree, for which a polynomial time algorithm is known (Luks =-=[41]-=-). Problem 21 can polynomially be reduced to this problem. For polytopes of bounded dimension both problems are polynomial time equivalent. Related problems: 21, 20s23. Selfduality of Polytopes Some A... |

132 |
Theory of linear and integer programming, Wiley-Interscience series in discrete mathematics
- Schrijver
- 1986
(Show Context)
Citation Context ...is case the numbers in the second description can be chosen such that their coding lengths depend only polynomially on the coding lengths of the numbers in the first description (see, e.g., Schrijver =-=[55]-=-). In our context, H- and V-descriptions are usually meant to be rational. By linear programming, each type of description can be made non-redundant in polynomial time (though it is unknown whether th... |

131 |
Munkres, Elements of Algebraic Topology
- R
- 1984
(Show Context)
Citation Context ...isted are direct generalizations of problems on polytopes. Most of the basic notions relevant in our context can be looked up in [65]; for topological concepts like homology we refer to Munkres’ book =-=[48]-=-. A finite abstract simplicial complex ∆ is a non-empty set of subsets (the simplices or faces) of a finite set of vertices such that F ∈ ∆ and G ⊂ F implysSome Algorithmic Problems in Polytope Theory... |

116 | A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant
- Cryan, Dyer
(Show Context)
Citation Context ...PRAS Status (fixed dim.): Polynomial time Some Algorithmic Problems in Polytope Theory 9 Dyer and Frieze [15] showed that the general problem is #P-hard (and #Peasy as well). Dyer, Frieze, and Kannan =-=[16]-=- found a Fully Polynomial Randomized Approximation Scheme (FPRAS) for the problem, i.e., a family (Aε)ε>0 of randomized algorithms, where, for each ε > 0, Aε computes a number Vε with the property tha... |

95 | Polymake: a framework for analyzing convex polytopes
- Gawrilow, M
- 2000
(Show Context)
Citation Context ...reated as finite objects by definition. This makes it possible to investigate (the smaller ones among) them by computer programs (like the polymake-system written by Gawrilow and Joswig, see [26] and =-=[27,28]-=-). Once chosen to exploit this possibility, one immediately finds oneself confronted with many algorithmic challenges. This paper contains descriptions of 35 algorithmic problems about polyhedra. The ... |

92 |
A subexponential randomized simplex algorithm
- Kalai
- 1992
(Show Context)
Citation Context ... known. In particular, no polynomial time variant of the simplex algorithm is known. However, a randomized version of the simplex algorithm solves the problem in (expected) subexponential time (Kalai =-=[36]-=-, Matouˇsek, Sharir, and Welzl [42]). Related problems: 25, 26, 27s16 Volker Kaibel and Marc E. Pfetsch 25. Optimal Vertex Input: H-description of a polyhedron P ⊂Éd , c ∈Éd Output: inf � c T v | v ve... |

87 |
Automorphism groups, isomorphism, reconstruction
- Babai
- 1996
(Show Context)
Citation Context ...empts to find a polynomial time algorithm for it have failed so far. Actually, the same holds for a lot of problems that can be polynomially reduced to the graph isomorphism problem (see, e.g., Babai =-=[3]-=-). 19. Affine Equivalence of V-Polytopes Input: Two polytopes P and Q given in V-description Output: “Yes” if P is affinely equivalent to Q, “No” otherwise Status (general): Graph isomorphism hard Sta... |

83 | How good are convex hull algorithms
- Avis, Bremner, et al.
- 1997
(Show Context)
Citation Context ... input (e.g., Cartesian products of suitably chosen two-dimensional polytopes and prisms over them). Vertex Enumeration is strongly polynomially equivalent to Problem 3 (see Avis, Bremner, and Seidel =-=[1]-=-). Since Problem 2 is strongly polynomially equivalent to Problem 3 as well, Vertex Enumeration is also strongly polynomially equivalent to Problem 2. For fixed d, Chazelle [12] found an O � m ⌊d/2⌋� ... |

80 |
Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie. Reprint der 1934 Auflage. Grundlehren der Mathematischen Wissenschaften
- Steinitz, Rademacher
- 1976
(Show Context)
Citation Context ...trical data is discussed in Section 2. The problems listed in this section are among the first ones asked in (modern) polytope theory, going back to the work of Steinitz and Radermacher in the 1930’s =-=[59]-=-. 29. Steinitz Problem Input: Lattice L Output: “Yes” if L is isomorphic to the face lattice of a polytope, “No” otherwise Status (general): N P-hard Status (fixed dim.): N P-hard If L is isomorphic t... |

76 |
An optimal convex hull algorithm in any fixed dimension
- Chazelle
- 1993
(Show Context)
Citation Context ...is, Bremner, and Seidel [1]). Since Problem 2 is strongly polynomially equivalent to Problem 3 as well, Vertex Enumeration is also strongly polynomially equivalent to Problem 2. For fixed d, Chazelle =-=[12]-=- found an O � m ⌊d/2⌋� polynomial time algorithm, which is optimal by the Upper Bound Theorem of McMullen [43]. There exist algorithms which are faster than Chazelle’s algorithm for small n, e.g., an ... |

70 |
Constructing higher-dimensional convex bulls at logarithmic cost per face
- Seidel
- 1980
(Show Context)
Citation Context ...ace lattice of the polytope. Swart [60], analyzing an algorithm of Chand and Kapur [10], proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel =-=[56]-=-. Fukuda, Liebling, and Margot [22] gave an algorithm which uses working space (without space for the output) bounded polynomially in the input size, but it has to solve many linear programs. For fixe... |

66 | Output-sensitive results on convex hulls, extreme points, and related problems, Discrete Comput
- Chan
- 1996
(Show Context)
Citation Context ...mal by the Upper Bound Theorem of McMullen [43]. There exist algorithms which are faster than Chazelle’s algorithm for small n, e.g., an O � m log n + (mn) 1−1/(⌊d/2⌋+1) polylog m � algorithm of Chan =-=[9]-=-. For general d, the reverse search method of Avis and Fukuda [2] solves the problem for simple polyhedra in polynomial total time, using working spacesSome Algorithmic Problems in Polytope Theory 5 (... |

65 |
The complexity of computing the volume of a polyhedron
- Dyer, Frieze
- 1988
(Show Context)
Citation Context ... Volume Input: Polytope P in H-description Output: The volume of P Status (general): #P-hard, FPRAS Status (fixed dim.): Polynomial time Some Algorithmic Problems in Polytope Theory 9 Dyer and Frieze =-=[15]-=- showed that the general problem is #P-hard (and #Peasy as well). Dyer, Frieze, and Kannan [16] found a Fully Polynomial Randomized Approximation Scheme (FPRAS) for the problem, i.e., a family (Aε)ε>0... |

64 |
The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, in “Topology and geometry - Rohlin seminar
- Mnëv
- 1988
(Show Context)
Citation Context ...d in the plane (linear time [30,45]) and check for 3-connectedness (in linear time, see Hopcroft and Tarjan [29]). Mnëv proved that the Steinitz Problem for d-polytopes with d +4 vertices is N P-hard =-=[47]-=-. Even more, Richter-Gebert [53] proved that for (fixed) d ≥ 4 the problem is N P-hard. For fixed d ≥ 4 it is neither known whether the problem is in N P nor whether it is in coN P. It seems unlikely ... |

60 |
An algorithm for convex polytopes
- Chand, Kapur
- 1970
(Show Context)
Citation Context ...nomial time See comments on Problem 1. Many algorithms for the Vertex Enumeration Problem in fact compute the whole face lattice of the polytope. Swart [60], analyzing an algorithm of Chand and Kapur =-=[10]-=-, proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel [56]. Fukuda, Liebling, and Margot [22] gave an algorithm which uses working space (wit... |

43 |
Computers and intractability: a guide to the Theory of N P-Completeness. Freeman and Company
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...e use without explanation we refer to Ziegler’s book [65]. Similarly, for the concepts from the theory of computational complexity that play a role here we refer to Garey and Johnson’s classical text =-=[24]-=-. Whenever we talk about polynomial reductions this refers to polynomial time Turing-reductions. For some of the problemssSome Algorithmic Problems in Polytope Theory 3 the output can be exponentially... |

42 |
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
- 1986
(Show Context)
Citation Context ...ed dim.): Polynomial time There exists a polynomial time algorithm if ∆ is given by the list of all simplices, since the Smith normal form of an integer matrix can be computed efficiently (Iliopoulos =-=[31]-=-). For fixed i or dim(∆)−i, the sizes of the boundary matrices are polynomial in the size of ∆ and the Smith normal form can again be computed efficiently. Related problems: 31, 32s34. Shellability So... |

38 |
A simple way to tell a simple polytope from its graph
- Kalai
- 1988
(Show Context)
Citation Context ...tion orientation (AOF-orientation) if it is acyclic. General US-orientations of graphs of cubes have recently received some attention (Szabó and Welzl [61]). AOForientations were used, e.g., by Kalai =-=[35]-=-. Since their linear extensions are precisely the shelling orders of the dual polytope, they have been considered much earlier. 13. Face Lattice of Combinatorial Polytopes Input: Vertex-facet incidenc... |

35 |
enumeration problems in geometry and combinatorics
- Linial, Hard
- 1986
(Show Context)
Citation Context ...Related problems: 1, 5 7. Number of Vertices Input: Polytope P in H-description Output: Number of vertices of P Status (general): #P-complete Status (fixed dim.): Polynomial time Dyer [14] and Linial =-=[40]-=- independently proved that Number of Vertices is #P-complete. It follows that the problem of computing the f-vector of P is #P-hard. Furthermore, Dyer [14] proved that the decision version (“Given a n... |

34 | Primal-dual methods for vertex and facet enumeration
- Bremner, Fukuda, et al.
- 1998
(Show Context)
Citation Context ...ynomial total time, using working spacesSome Algorithmic Problems in Polytope Theory 5 (without space for output) bounded polynomially in the input size. An algorithm of Bremner, Fukuda, and Marzetta =-=[8]-=- solves the problem for simplicial polytopes. Note that these algorithms need a vertex of P to start from. Provan [52] gives a polynomial total time algorithm for enumerating the vertices of polyhedra... |

34 | On the embedding phase of the Hopcroft and Tarjan planarity testing algorithm. Algorithmica
- Mehlhorn, Mutzel
- 1996
(Show Context)
Citation Context .... The problem can be solved in polynomial time in dimension at most three by computing a planar embedding of the graph, which can be done in linear time (Hopcroft and Tarjan [30], Mehlhorn and Mutzel =-=[45]-=-). Related problems: 16, 17, 18 16. Facet System Verification for Simple Polytopes Input: The (abstract) graph GP of a simple polytope P and a family F of subsets of nodes of GP Output: “Yes” if F is ... |

31 |
Finding the convex hull facet by facet
- Swart
- 1985
(Show Context)
Citation Context ... Polynomial total time Status (fixed dim.): Polynomial time See comments on Problem 1. Many algorithms for the Vertex Enumeration Problem in fact compute the whole face lattice of the polytope. Swart =-=[60]-=-, analyzing an algorithm of Chand and Kapur [10], proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel [56]. Fukuda, Liebling, and Margot [22]... |

30 |
The complexity of vertex enumeration methods
- Dyer
- 1983
(Show Context)
Citation Context ...P-complete Status (fixed dim.): Polynomial timesSome Algorithmic Problems in Polytope Theory 7 Independently proved to be N P-complete in the papers of Chandrasekaran, Kabadi, and Murty [11] and Dyer =-=[14]-=-. Fukuda, Liebling, and Margot [22] proved that the problem is strongly N P-complete. For fixed dimension, one can enumerate all vertices in polynomial time (see Problem 1) and check whether they are ... |

27 | New lower bounds for convex hull problems in odd dimensions
- Erickson
- 1999
(Show Context)
Citation Context ...Fukuda, and Marzetta [8] noted that if P is given in V-description the problem is polynomial time solvable: enumerate the edges (1-skeleton, see Problem 5) and apply the Lower Bound Theorem. Erickson =-=[19]-=- showed that in the worst case Ω(m ⌈d/2⌉−1 + m log m) sideness queries are required to test whether a polytope is simple. For odd d this matches the upper bound. A sideness query is a question of the ... |

27 |
Algorithmen zur formalen Begriffsanalyse, in Beiträge zur Begriffsanalyse
- Ganter
- 1987
(Show Context)
Citation Context ...n compute χ(∆) in time O � |V| 2� by a Möbius function approach, see Rota [54]. Usually V is much smaller than the whole face lattice of ∆. V can be listed lexicographically by an algorithm of Ganter =-=[23]-=-, in time O(min{m, n} · α · |V|), where α is the number of vertex-facets incidences. Related problems: 32 32. f-Vector of Simplicial Complexes Input: Finite abstract simplicial complex ∆ given by a li... |

25 |
Mani–Levitska: On puzzles and polytope isomorphisms
- Blind, P
- 1987
(Show Context)
Citation Context ...t) graph GP of a simple polytope P Output: The family of the subsets of nodes of GP corresponding to the vertex sets of the facets of P Status (general): Open Status (fixed dim.): Open Blind and Mani =-=[6]-=- proved that the entire combinatorial structure of a simple polytope is determined by its graph. This is false for general polytopes (of dimension at least four), which is the main reason why we restr... |

18 |
On the complexity of four polyhedral set containment problems, Mathematical Programming 33(2
- Freund, Orlin
- 1985
(Show Context)
Citation Context ...put: Polytope P given in H-description, polytope Q given in V-description Output: “Yes” if P ⊆ Q, “No” otherwise Status (general): coN P-complete Status (fixed dim.): Polynomial time Freund and Orlin =-=[20]-=- proved that this problem is coN P-complete. Note that the reverse question whether Q ⊆ P is trivial. The questions where either both P and Q are given in H-description or both in V-description can be... |

17 |
How to Draw a Graph,” Proc
- Tutte
- 1963
(Show Context)
Citation Context ...n dimensions one and two the problem is trivial. For a three-dimensional polytope P the problem can be solved in polynomial time, e.g., by producing a plane drawing of GP with convex faces (see Tutte =-=[62]-=-) and sorting the nodes with respect to a linear function (in general position). A polynomial time algorithm would lead to a polynomial algorithm for Problem 16 (see [32]). By duality of polytopes, th... |

15 |
Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron
- Fukuda, Liebling, et al.
- 1997
(Show Context)
Citation Context ...[60], analyzing an algorithm of Chand and Kapur [10], proved that there exists a polynomial total time algorithm for this problem. For a faster algorithm see Seidel [56]. Fukuda, Liebling, and Margot =-=[22]-=- gave an algorithm which uses working space (without space for the output) bounded polynomially in the input size, but it has to solve many linear programs. For fixed dimension, the size of the output... |

15 |
K.G.: Some NP-complete problems in linear programming
- Chandrasekaran, Kaboadi, et al.
- 1982
(Show Context)
Citation Context ...ral): Strongly NP-complete Status (xed dim.): Polynomial time Some Algorithmic Problems in Polytope Theory 7 Independently proved to be NP-complete in the papers of Chandrasekaran, Kabadi, and Murty [=-=11]-=- and Dyer [14]. Fukuda, Liebling, and Margot [22] proved that the problem is strongly NP-complete. Forsxed dimension, one can enumerate all vertices in polynomial time (see Problem 1) and check whethe... |

13 |
Efficient enumeration of the vertices of polyhedra associated with network LP’s
- Provan
- 1994
(Show Context)
Citation Context ...nded polynomially in the input size. An algorithm of Bremner, Fukuda, and Marzetta [8] solves the problem for simplicial polytopes. Note that these algorithms need a vertex of P to start from. Provan =-=[52]-=- gives a polynomial total time algorithm for enumerating the vertices of polyhedra arising from networks. There are many more algorithms known for this problem – none of them is a polynomial total tim... |

10 |
A representation of 2-dimensional pseudomanifolds and its use in the design of a linear time shelling algorithm
- Danaraj, Klee
- 1978
(Show Context)
Citation Context ... it is unclear if the problem can be solved in polynomial time if ∆ is given by a list of all simplices. For two-dimensional pseudo-manifolds the problem can be solved in linear time (Danarj and Klee =-=[13]-=-). Related problems: 17, 35 35. Partitionability Input: Finite abstract simplicial complex ∆ given by a list of facets Output: “Yes” if ∆ is partionable, “No” otherwise Status (general): Open Status (... |

10 |
Elementare Theorie der konvexen Polyeder
- Weyl
- 1935
(Show Context)
Citation Context ... time” only makes sense with respect to problems which explicitly require the output to be non-redundant. A very fundamental result in the theory of convex polyhedra is due to Minkowski [46] and Weyl =-=[64]-=-. For the special case of polytopes (to which we restrict our attention from now on) it can be formulated as follows. Every polytope P ⊂Êd can be specified by an H- or by a V-description. Here, an H-d... |

9 | The random-facet simplex algorithm on combinatorial cubes
- Gärtner
(Show Context)
Citation Context ... problem can be solved in a subexponential number of oracle calls by the random facet variant of the simplex algorithm due to Kalai [36]. For a derivation of the explicit bound e 2√ d − 1 see Gärtner =-=[25]-=-. In fixed dimension the problem is trivial by mere enumeration. The problem generalizes linear programming problems whose sets of feasible solutions are combinatorially equivalent to cubes. Related p... |

9 | On the k-systems of a simple polytope
- Joswig, Kaibel, et al.
(Show Context)
Citation Context ..., and constructive proof of Blind and Mani’s result. However, the algorithm that can be derived from it has a worst-case running time that is exponential in the number of vertices of the polytope. In =-=[32]-=- it is shown that the problem can be formulated as a combinatorial optimization problem for which the problem to find an AOF-orientation of GP (see Problem 17) is strongly dual in the sense of combina... |

9 | Computing the face lattice of a polytope from its vertex-facet incidences - Kaibel, Pfetsch - 2002 |

9 | Signable posets and partitionable simplicial complexes
- Kleinschmidt, Onn
- 1996
(Show Context)
Citation Context ...ber of faces of ∆?” is contained in N P.s20 Volker Kaibel and Marc E. Pfetsch This problem is only known to be in N P for partitionable (see Problem 18) simplicial complexes (see Kleinschmidt and Onn =-=[39]-=-). To the best of our knowledge, no proof of #P-hardness of the general problem has appeared in the literature. Therefore we include one here. Consider an instance of SAT, i.e., a formula in conjuncti... |

6 | The complexity of finding small triangulations of convex 3-polytopes
- Below, Loera, et al.
- 2004
(Show Context)
Citation Context ...a (pure) ddimensional geometric simplicial complex (see Section 7). The size of T is the number of its d-simplices. Every (convex) polytope admits a triangulation. Below, De Loera, and Richter-Gebert =-=[4,5]-=- proved that Minimum Triangulation is N P-complete for (fixed) d ≥ 3. Furthermore, it is N P-hard to compute a triangulation of minimal size for (fixed) d ≥ 3.s12. Volume Input: Polytope P in H-descri... |

4 |
Optimal scaling of balls and polyhedra
- Eaves, Freund
- 1982
(Show Context)
Citation Context ...e question whether Q ⊆ P is trivial. The questions where either both P and Q are given in H-description or both in V-description can be solved by linear programming (Problem 24), see Eaves and Freund =-=[17]-=-. For fixed dimension, one can enumerate all vertices of P in polynomial time (see Problem 1) and compare the descriptions of P and Q (after removing redundant points). Related problems: 3 5. Face Lat... |