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Generating All Vertices of a Polyhedron Is Hard

2007
*
Discrete & Computational Geometry
*

Yet, for

doi:10.1007/s00454-006-1259-6
fatcat:4oausme7ejfolhbegnti222y74
*generating*maximal feasible subsystems the complexity remains open. (ii) Given*a*feasible system*of*linear inequalities,*generating**all**vertices**of*the corresponding*polyhedron**is**hard*. ... We show that*generating**all*negative cycles*of**a*weighted graph*is**a**hard*enumeration problem, in both the directed and undirected cases. ...*A*further consequence*of*Theorem 1*is*that enumerating*all**vertices**of**a*bounded*polyhedron*P which do not belong to*a*given face*of*P*is*also*hard*, in*general*. ...##
###
Generating All Vertices of a Polyhedron Is Hard

2008
*
Discrete & Computational Geometry
*

Yet, for

doi:10.1007/s00454-008-9050-5
fatcat:s5been3h6bcnthbkmvkbhks3m4
*generating*maximal feasible subsystems the complexity remains open. (ii) Given*a*feasible system*of*linear inequalities,*generating**all**vertices**of*the corresponding*polyhedron**is**hard*. ... We show that*generating**all*negative cycles*of**a*weighted graph*is**a**hard*enumeration problem, in both the directed and undirected cases. ...*A*further consequence*of*Theorem 1*is*that enumerating*all**vertices**of**a*bounded*polyhedron*P which do not belong to*a*given face*of*P*is*also*hard*, in*general*. ...##
###
Generating all vertices of a polyhedron is hard

2006
*
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
*

Yet, for

doi:10.1145/1109557.1109640
fatcat:tbmwzijiq5codeswcznstn4acy
*generating*maximal feasible subsystems the complexity remains open. (ii) Given*a*(feasible) system*of*linear inequalities,*generating**all**vertices**of*the corresponding*polyhedron**is**hard*. ... We show that*generating**all*negative cycles*of**a*weighted graph*is**a**hard*enumeration problem, in both the directed and undirected cases. ... We shall derive several consequences*of*the above results, including the*hardness**of**generating**all**vertices**of**a*(possibly unbounded)*polyhedron*,*generating**all*minimal infeasible subsystems*of**a*system ...##
###
Generating All Vertices of a Polyhedron Is Hard
[chapter]

*
Twentieth Anniversary Volume:
*

Yet, for

doi:10.1007/978-0-387-87363-3_17
fatcat:ojcqqnaa5nb6hlprep7ynvkrju
*generating*maximal feasible subsystems the complexity remains open. (ii) Given*a*(feasible) system*of*linear inequalities,*generating**all**vertices**of*the corresponding*polyhedron**is**hard*. ... We show that*generating**all*negative cycles*of**a*weighted graph*is**a**hard*enumeration problem, in both the directed and undirected cases. ... We shall derive several consequences*of*the above results, including the*hardness**of**generating**all**vertices**of**a*(possibly unbounded)*polyhedron*,*generating**all*minimal infeasible subsystems*of**a*system ...##
###
The negative cycles polyhedron and hardness of checking some polyhedral properties

2010
*
Annals of Operations Research
*

Based on this characterization, and using

doi:10.1007/s10479-010-0690-5
fatcat:jx3qytsbwba3zfithaoxywnjse
*a*construction developed in [11], we show that, unless P = N P , there*is*no output polynomial-time algorithm to*generate**all*the*vertices**of**a*0/1-*polyhedron*. ... Finally, we also show that it*is*NP-*hard*to approximate the maximum support*of**a*vertex*a**polyhedron*in R n within*a*factor*of*12/n. ... The latter problem*is*NP-*hard*by Lemma 2. Thus, it*is*NP-*hard*to*generate**all**vertices**of**a*0/1-*polyhedron*. ...##
###
Characterization of the Vertices and Extreme Directions of the Negative Cycles Polyhedron and Hardness of Generating Vertices of 0/1-Polyhedra
[article]

2008
*
arXiv
*
pre-print

As

arXiv:0801.3790v2
fatcat:skpo5ybpbzdq7jwe3ljn2bxaf4
*a*corollary, we show that, unless P=NP, there*is*no output polynomial-time algorithm to*generate**all*the*vertices**of**a*0/1-*polyhedron*. ... Given*a*graph G=(V,E) and*a*weight function on the edges w:E, we consider the*polyhedron*P(G,w)*of*negative-weight flows on G, and get*a*complete characterization*of*the*vertices*and extreme directions ... Thus, it*is*NP-*hard*to*generate**all**vertices**of**a*0/1-*polyhedron*. ...##
###
When can a graph form an orthogonal polyhedron?

2004
*
Canadian Conference on Computational Geometry
*

Our construction uses

dblp:conf/cccg/BiedlG04
fatcat:obotwkctardwvaq4wx5rw2mwcq
*a**polyhedron**of*genus 1, but can be*generalized*to polyhedra*of*genus 0. The reduction*is*from PARTITION, which*is*known to be NP-*hard*[6] . ... The NP-*hardness*proof requires given edge lengths.*Is*it NP-*hard*to test whether*a*given graph without edge lengths*is*the graph*of*some orthogonal*polyhedron*? ...##
###
On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

2008
*
Discrete & Computational Geometry
*

For the Minkowski sum, we prove that enumerating the facets

doi:10.1007/s00454-008-9097-3
fatcat:7nrq464d3ffsrgp5y7l36ludbu
*of*P 1 + P 2*is*NP-*hard*if P 1 and P 2 are specified by facets, or if P 1*is*specified by*vertices*and P 2*is**a*polyhedral cone specified by ... Also, computing the*vertices**of*the intersection*of*two polytopes given by*vertices**is*shown to be NP-*hard*. ... Acknowledgements The author was supported by Graduiertenkolleg fellowship for PhD studies provided by Deutsche Forschungsgemeinschaft when some*of*this research was done. ...##
###
Face-Guarding Polyhedra
[article]

2014
*
arXiv
*
pre-print

Then we show that it

arXiv:1305.2866v4
fatcat:ihb7recnfnd7lgnrhppaqo3tdy
*is*NP-*hard*to approximate the minimum number*of*(closed or open) face guards within*a*factor*of*Omega(log f), even for polyhedra that are orthogonal and simply connected. ... Along the way we discuss some applications, arguing that face guards are not*a*reasonable model for guards patrolling on the surface*of**a**polyhedron*. ... Acknowledgments The author wishes to thank the anonymous reviewers for precious suggestions on how to improve the readability*of*this paper. ...##
###
3D Building Model Fitting Using A New Kinetic Framework
[article]

2008
*
arXiv
*
pre-print

We describe

arXiv:0805.0648v1
fatcat:tkwc4uapmfgghmz6m65732bpfq
*a*new approach to fit the*polyhedron*describing*a*3D building model to the point cloud*of**a*Digital Elevation Model (DEM). ... This new kinetic framework allows the manipulation*of**a*bounded*polyhedron*with simple faces by specifying the target plane equations*of*each*of*its faces. ... (n · n) w 0 .w 1 .w 2 (6) If*all*the 3 edges*of*the triangle are soft (triangulation) edges and not*hard*edges*of*the*polyhedron*(Fig. 6) , the λ polynom*of*the numerator*of*the certificate cannot*generally*...##
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On the hardness of minkowski addition and related operations

2007
*
Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07
*

Since the convex hull

doi:10.1145/1247069.1247124
dblp:conf/compgeom/Tiwary07
fatcat:f6x4sofafbazvilofhf73rj5lm
*of*the union and the intersection*of*two polytopes relate naturally to the Minkowski sum via the Cayley trick and polarity, similar*hardness*results follow for these operations as ... In particular, this shows that there*is*no output sensitive polynomial algorithm to compute the facets*of*the Minkowski sum*of*two arbitrary H-polytopes even if we consider only rational polytopes. ... It was shown by Khachiyan et. al [13] that it*is*coNP-*Hard*to enumerate*all**vertices**of**a**polyhedron*given by its facets. The following theorem restates the result*of*[13] . THEOREM 2. ...##
###
Generating vertices of polyhedra and related problems of monotone generation
[chapter]

2009
*
CRM Proceedings and Lecture notes AMS
*

The well-known vertex enumeration problem calls for

doi:10.1090/crmp/048/02
fatcat:ucvd4twcsvdtnbdihuyzlnaxdq
*generating**all**vertices**of**a**polyhedron*, given by its description as*a*system*of*linear inequalities. ... We also discuss their limitations and sketch an NP-*hardness*proof for*generating*the*vertices**of**general*polyhedra. ... Given*a**polyhedron*P by (1.1), problem Gen(P )*of**generating**all**vertices**of*P*is*NP-*hard*. ...##
###
When can a net fold to a polyhedron?

2005
*
Computational geometry
*

In this paper, we study the problem

doi:10.1016/j.comgeo.2004.12.004
fatcat:o26nlhidbzaizecqcfu2x2qxdq
*of*whether*a**polyhedron*can be obtained from*a*net by folding along the creases. ... We show that this problem can be solved in polynomial time if the dihedral angle at each crease*is*given, and it becomes NP-*hard*if these angles are unknown. ... Acknowledgements We thank an anonymous referee for finding an error in the NP-*hardness*reduction. ...##
###
Searching Polyhedra by Rotating Half-Planes
[article]

2011
*
arXiv
*
pre-print

Finally we show that deciding whether

arXiv:1104.4137v5
fatcat:5lszplzt2bb3xnbnaaifxnh2nu
*a*given set*of*guards has*a*successful search schedule*is*strongly NP-*hard*, and that deciding if*a*given target area*is*searchable at*all**is*strongly PSPACE-*hard*, ... (Minimizing the number*of*guards to search*a*given*polyhedron**is*easily seen to be NP-*hard*.) ... This work was supported in part by MIUR*of*Italy under project Algo-DEEP prot. 2008TFBWL4. ...##
###
Face-guarding polyhedra

2014
*
Computational geometry
*

Then we show that it

doi:10.1016/j.comgeo.2014.04.009
fatcat:h3dhgq4wqngtplfmg47fbsafiq
*is*NP-*hard*to approximate the minimum number*of*(closed or open) face guards within*a*factor*of*Ω(log f ), even for polyhedra that are orthogonal and simply connected. ... Along the way we discuss some applications, arguing that face guards are not*a*reasonable model for guards patrolling on the surface*of**a**polyhedron*. ... It was proved in [1] that Set Cover*is*NP-*hard*to approximate within*a*ratio*of*Ω(log n) and, by inspecting the reduction employed, it*is*apparent that*all*the*hard*Set Cover instances*generated*are ...
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