In physics, the no-signaling approach to quantum nonlocality [ 2 — 4 , 7 ] leads to degenerate systems of linear constraints for the correlations between the parties playing in a correlational set up [ 4 , 6 , 7 ]. In systems biology, the structural analysis approach for the mathematical modeling of a biological complex applies constraint-based methods that take the form of degenerate linear systems of inequalities [ 8 — 11 ]. In order to transform the model constraints into quantitative predictions, conversion methods are necessary to derive the relational structure a skeleton graph or a network that is conformed by the set of extreme points of the feasible region.
In the no-signaling approach to quantum nonlocality, the extreme points of the polytope of correlations provide the local operations and the elementary non-local Popescu-Rohrlich channels [ 2 — 4 ] which are necessary and in any cases sufficient to simulate any no-signaling correlational scenario [ 5 ]. In both cases, the resulting output descriptions display a wide range of structural complexities, characterized by measurements as graph entropy [ 13 , 14 ] and graph similarity [ 15 ].
The most effective conversion methods available arise from combinatorial geometry [ 16 — 19 ]. However, when the half-space descriptions of the extreme points of the feasible polytope are degenerate a combinatorial explosion is produced that is the cause of stubborn difficulties for their enumeration [ 19 — 23 ]. The paper introduces an algorithm to convert a half-space description into the skeleton graph conformed by the set of extreme points of the feasible region.
Using a combination of incremental [ 18 , 19 ] and pivotal [ 24 , 25 ] methods, an algorithm with a good performance to resolve degeneracy and to complete the traversal of the set of extreme points is presented. The effects of degeneracy are studied computationally for a very large number of half-space descriptions, organized into four families according to the degree of degeneracy of the input half-space descriptions and on the complexity of the output graphs. The model constraints determine the entries of A and the entries of vector.
Each one of the m constraint inequalities in Eq 1 defines a half-space of the euclidean space of dimension n , E n.
Every row of matrix A is the constraint vector , with index. Vector a i defines the i -th feasible half-space. The intersection of all half-spaces constitutes the set of feasible values of , which conform the convex polytope. The description of P that is provided by Eq 1 is known as a half-space description, or H -description. However, what is physically meaningful is the combinatorial structure that is encoded in the skeleton graph G P of P , known as the V -description of P.
First, a point p is in the feasible region P whenever the slack vector s p is non-negative. An extreme point p is regular or non-degenerate if it has an active set of cardianlity. The active set of a regular extreme point p , , is a basis. The combinatorial triviality of regular extreme points does not present any difficulty to the simplex pivoting rules [ 16 , 24 , 25 ].
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The amount of searches Eq 2 may be—per vertex! Besides, when doing the search, the simplex method may get trapped in an endless cycle or just get stalled [ 22 ]. Several approaches have been tried out to overcome such deficiencies [ 20 — 23 ]. The practice of the double description method has proved its efficiency [ 18 , 19 ] in the determination of the extreme rays of highly degenerate polyhedral cones [ 26 ]. However, the method is not efficient for the construction of the complete skeleton graph of large degenerate systems of linear inequalities, mainly due to the very large number of tentative vertices that are produced at intermediate stages of the conversion procedure [ 26 ].
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The majority of intermediate vertices are discarded at the end. To overcome this situation, we have designed a swift and compact pivoting method to determine the neighbors of extreme points, by taking as the input the extreme rays of a cone.
In this way we have combined in Algorithm 5 the best of two methods: the incremental slicing of cones to defeat degeneracy [ 18 , 19 ], and pivoting around extreme points for a swift traversal of extreme points [ 24 ]. The incremental procedure goes slicing a cone, starting with a regular cone that is broader than and includes the target cone. The preparation of the base cone to be the input of the incremental procedure is explained in Section 1: a basis is chosen from the active set and a standard algebraic method produces the extreme rays of the corresponding regular cone.
The half-spaces in the active set that are not part of the basis set are inserted by the incremental procedure, one-by-one, until they are exhausted and the target cone has been sculpted. The half-space insertion procedure is explained in Section 2. The explanation includes i the alternative combinatorial or algebraic test necessary to identify the 2-face cuts during the slicing procedure and ii a recording strategy that helps the algorithm to reduce the number of tests.
The extreme rays that are produced by the incremental procedure provide the scanning directions for the pivoting rule that is followed to determine the set of neighboring points of p.
The pivoting rule is developed in Section 3. In Section 4 the incremental and pivoting methods are combined in Algorithm 5, which converts the system of linear inequalities into the skeleton graph conformed by the set of extreme points of the feasible region. The computational practice in Section 5 affords understanding about the effects that the degeneracy present in the input systems has on the performance of Algorithm 5. The very large number of input systems employed in the practice of Section 5 is organized in four families that offer a controlled and distributed sampling of the complexity spectrum of the conversion problem.
The family with the lowest complexity consists of regular non degenerate polytopes with H -descriptions produced at random [ 7 ]. The other three are one-parameter families. The family with the highest complexity consists of Birkhoff polytopes [ 27 , 28 ]. The other two families, with intermediate complexities, consist of no-signaling polytopes [ 7 ]. Section 5 details the characterization of the four families, produced by applying Algorithm 5.
The computational practice of Section 5 distinguishes two classes of conversion problems. A first class consists of systems of linear inequalities that have a combined complexity which becomes smaller as a function of the input size.assaithebourdesk.gq
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The systems in this class convert into skeleton graphs with a number of vertices that grows faster than their vertex-connectivity, as a function of the input size. For these conversion problems I the CPU time consumed by Algorithm 5 is mostly applied to complete the traversal of extreme points and not to resolve degeneracy, II the algebraic test for 2-face cuts in Algorithm 5 is faster than the combinatorial test and III the incremental procedure is highly sensitive i to the choice of the input basis set and ii to the insertion order of the cutting half-spaces. The second class of conversion problems distinguished by the computational practice in Section 5 has a combined complexity that does not decrease as a function of the input size.
As the input size of the systems of linear inequalities in this class is increased, the number of vertices of their skeleton graphs increases and the vertex-connectivity does not decrease.
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For these conversion problems I the CPU time consumed by Algorithm 5 is mostly applied to resolve the degeneracy present at the H -description and not to complete the traversal of extreme points, II the combinatorial test for 2-face cuts in Algorithm 5 is faster than the algebraic test and III the incremental procedure is not sensitive i to the choice of the input basis set and ii neither to the insertion order of the cutting half-spaces.
For the benefit of the implementor we make use of a simple, but rich enough, no-signaling constraint H -description [ 7 ] as example to illustrate our exposition. The conversion problem consists of the two-party correlations that are feasible for a no-signaling box [ 4 ] with a binary input per party and asymmetric in its outputs, producing one out of 3 and of 2 possible outcomes per party respectively. The no-signaling and non-negativity constraints on correlations [ 7 ] produce the system of linear inequalities in Table 1. The cone described by the active set of the extreme point p is the set.
The cone is the translation of cone to the origin, 3 Since rank , both cones and are peaked, with apices located at p and the origin, respectively. The polyhedral cone fits the feasible region P and the 1-faces or extreme rays of provide the directions to scan for the neighbors of p. Then, and in view of Eq 3 , the first step towards the skeleton graph G P is to convert the half-space description of cone into its set of extreme rays , such that. The determination of the set of extreme rays of a degenerate cone is the subject matter of the next section.
Meanwhile, the extreme rays of a regular cone may be obtained from its half-space description by methods of linear algebra. A regular cone is the intersection of the half-spaces of a basis. The set of extreme rays of is given in the following. Lemma 1. Let p be an extreme point of P. Let be a basis. The set of extreme rays of the cone is 4. The set in Lemma 1. Given that we have that. The set of extreme rays of cone is the set. For a degenerate vertex p with active set the double description method, discussed in the next section, produces the set of extreme rays of cone by starting with the set of rays provided by Lemma 1.
The basis , through lemma 1. The incremental procedure to generate the set of extreme rays of the cone at a degenerate extreme point p , which is described by the set of active planes , starts with the approximate set , provided by lemma 1. The set is not empty by degeneracy.
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Then, the half-spaces remaining in are introduced one at a time. The insertion process produces a non-increasing chain of cones by eliminating the current extreme rays which are not in the feasible half-space introduced and adding the new extreme rays that are created by the half-space that has been added. When all hyperplanes in the degeneracy set have been inserted, the set of extreme rays , describing cone , is produced. Assume the insertion procedure has gone adding half-spaces from as far as to generate , for some.
Assume the procedure is to advance one step farther by adding the half-space H k , for some k remaining in. Then, by working on the current set of rays and the vector a k that is associated to H k , the procedure will produce the set of extreme rays that results from the insertion of the half-space H k.
This one-plane insertion procedure defines the function. The procedure represented by F begins with the partition of the current set of rays into three sets. A first set collects the rays that are within the feasible half-space H k , the set.
A second set is , which is the set of rays outside the feasible half-space H k. We have. Central to the incremental slicing procedure [ 7 , 19 ] is the following. Lemma 2. The collection of all such rays, 7 completes the set of extreme rays of ,. The pair constitutes a double description of the cone.
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