Filters








9,921 Hits in 5.1 sec

Generating All Vertices of a Polyhedron Is Hard

Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni, Vladimir Gurvich
2007 Discrete & Computational Geometry  
Yet, for 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.  ... 
doi:10.1007/s00454-006-1259-6 fatcat:4oausme7ejfolhbegnti222y74

Generating All Vertices of a Polyhedron Is Hard

Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni, Vladimir Gurvich
2008 Discrete & Computational Geometry  
Yet, for 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.  ... 
doi:10.1007/s00454-008-9050-5 fatcat:s5been3h6bcnthbkmvkbhks3m4

Generating all vertices of a polyhedron is hard

Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni, Vladimir Gurvich
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06  
Yet, for 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  ... 
doi:10.1145/1109557.1109640 fatcat:tbmwzijiq5codeswcznstn4acy

Generating All Vertices of a Polyhedron Is Hard [chapter]

Leonid Khachiyan, Endre Boros, Konrad Borys, Vladimir Gurvich, Khaled Elbassioni
Twentieth Anniversary Volume:  
Yet, for 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  ... 
doi:10.1007/978-0-387-87363-3_17 fatcat:ojcqqnaa5nb6hlprep7ynvkrju

The negative cycles polyhedron and hardness of checking some polyhedral properties

Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Hans Raj Tiwary
2010 Annals of Operations Research  
Based on this characterization, and using 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.  ... 
doi:10.1007/s10479-010-0690-5 fatcat:jx3qytsbwba3zfithaoxywnjse

Characterization of the Vertices and Extreme Directions of the Negative Cycles Polyhedron and Hardness of Generating Vertices of 0/1-Polyhedra [article]

Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Hans Raj Tiwary
2008 arXiv   pre-print
As 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.  ... 
arXiv:0801.3790v2 fatcat:skpo5ybpbzdq7jwe3ljn2bxaf4

When can a graph form an orthogonal polyhedron?

Therese C. Biedl, Burkay Genç
2004 Canadian Conference on Computational Geometry  
Our construction uses 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?  ... 
dblp:conf/cccg/BiedlG04 fatcat:obotwkctardwvaq4wx5rw2mwcq

On the Hardness of Computing Intersection, Union and Minkowski Sum of Polytopes

Hans Raj Tiwary
2008 Discrete & Computational Geometry  
For the Minkowski sum, we prove that enumerating the facets 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.  ... 
doi:10.1007/s00454-008-9097-3 fatcat:7nrq464d3ffsrgp5y7l36ludbu

Face-Guarding Polyhedra [article]

Giovanni Viglietta
2014 arXiv   pre-print
Then we show that it 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.  ... 
arXiv:1305.2866v4 fatcat:ihb7recnfnd7lgnrhppaqo3tdy

3D Building Model Fitting Using A New Kinetic Framework [article]

Mathieu Brédif, Dider Boldo, Marc Pierrot-Deseilligny, Henri Maître
2008 arXiv   pre-print
We describe 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  ... 
arXiv:0805.0648v1 fatcat:tkwc4uapmfgghmz6m65732bpfq

On the hardness of minkowski addition and related operations

Hans Raj Tiwary
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
Since the convex hull 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.  ... 
doi:10.1145/1247069.1247124 dblp:conf/compgeom/Tiwary07 fatcat:f6x4sofafbazvilofhf73rj5lm

Generating vertices of polyhedra and related problems of monotone generation [chapter]

Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Kazuhisa Makino
2009 CRM Proceedings and Lecture notes AMS  
The well-known vertex enumeration problem calls for 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.  ... 
doi:10.1090/crmp/048/02 fatcat:ucvd4twcsvdtnbdihuyzlnaxdq

When can a net fold to a polyhedron?

Therese Biedl, Anna Lubiw, Julie Sun
2005 Computational geometry  
In this paper, we study the problem 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.  ... 
doi:10.1016/j.comgeo.2004.12.004 fatcat:o26nlhidbzaizecqcfu2x2qxdq

Searching Polyhedra by Rotating Half-Planes [article]

Giovanni Viglietta
2011 arXiv   pre-print
Finally we show that deciding whether 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.  ... 
arXiv:1104.4137v5 fatcat:5lszplzt2bb3xnbnaaifxnh2nu

Face-guarding polyhedra

Giovanni Viglietta
2014 Computational geometry  
Then we show that it 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  ... 
doi:10.1016/j.comgeo.2014.04.009 fatcat:h3dhgq4wqngtplfmg47fbsafiq
« Previous Showing results 1 — 15 out of 9,921 results