Extreme Rays Research Articles

The expressive power of binary submodular functions

We investigate whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This question has been considered in several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular polynomials of arity 4 can be expressed by binary submodular polynomials. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish, for the first time, that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.

Comment on “Extreme rays in [formula omitted] entangled edge states with positive partial transposes” [Phys. Lett. A 369 (2007) 16

Recently, the author [K.-C. Ha, Phys. Lett. A 361 (2007) 515] found examples of 3 ⊗ 3 edge PPTES of type ( 5 , 5 ) and ( 6 , 6 ). Kim and Kye [W.C. Kim, S.-H. Kye, Phys. Lett. A 369 (2007) 16] proved that amongst these the ( 5 , 5 ) edge PPTES generate extreme rays in the cone of all positive semi-definite matrices with positive semi-definite partial transposes. Here we show that this is also the case for the ( 6 , 6 ) state.

The present status of the Telescope Array experiment

The Telescope Array(TA) experiment located at western desert in Utah USA (N39.3,W112.9) is designed for observation of air shower from extreme high energy cosmic rays. The TA detector consists of 2 types of detector to enable a cross check on systematic difference from the two main methods of observation for the energy region. One is a Fluorescence detector (FD) for detecting fluorescence light from air shower and another is surface detector (SD) array for detecting air shower particles at ground level. Each SD consists of 2 layers of plastic scintillator with 3m 2 of surface and more sensitive to electromagnetic component in air shower. The full operation using 3FD stations and full SD array has started. Here we present the updated status of Telescope Array experiment.

Mixing MIR inequalities with two divisible coefficients

This article is a polyhedral study of a generalization of the mixing set where two different, divisible coefficients are allowed for the integral variables. Our results generalize earlier work on mixed integer rounding, mixing, and extensions. They also directly apply to applications such as production planning problems involving lower bounds or start-ups on production, when these are modeled as mixed-integer linear programs. We define a new class of valid inequalities and give two proofs that they suffice to describe the convex hull of this mixed-integer set. We give a characterization of each of the maximal faces of the convex hull, as well as a closed form description of its extreme points and rays, and show how to separate over this set in $${\mathcal{O} (n\log n)}$$. Finally, we give several extended formulations of polynomial size, and study conditions under which adding certain simple constraints on the integer variables preserves our main result.

Determining maximal efficient faces in multiobjective linear programming problem

Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly.

Large-scale computation of elementary flux modes with bit pattern trees

Elementary flux modes (EFMs)--non-decomposable minimal pathways--are commonly accepted tools for metabolic network analysis under steady state conditions. Valid states of the network are linear superpositions of elementary modes shaping a polyhedral cone (the flux cone), which is a well-studied convex set in computational geometry. Computing EFMs is thus basically equivalent to extreme ray enumeration of polyhedral cones. This is a combinatorial problem with poorly scaling algorithms, preventing the large-scale analysis of metabolic networks so far. Here, we introduce new algorithmic concepts that enable large-scale computation of EFMs. Distinguishing extreme rays from normal (composite) vectors is one critical aspect of the algorithm. We present a new recursive enumeration strategy with bit pattern trees for adjacent rays--the ancestors of extreme rays--that is roughly one order of magnitude faster than previous methods. Additionally, we introduce a rank updating method that is particularly well suited for parallel computation and a residue arithmetic method for matrix rank computations, which circumvents potential numerical instability problems. Multi-core architectures of modern CPUs can be exploited for further performance improvements. The methods are applied to a central metabolism network of Escherichia coli, resulting in approximately 26 Mio. EFMs. Within the top 2% modes considering biomass production, most of the gain in flux variability is achieved. In addition, we compute approximately 5 Mio. EFMs for the production of non-essential amino acids for a genome-scale metabolic network of Helicobacter pylori. Only large-scale EFM analysis reveals the >85% of modes that generate several amino acids simultaneously. An implementation in Java, with integration into MATLAB and support of various input formats, including SBML, is available at http://www.csb.ethz.ch in the tools section; sources are available from the authors upon request.

A new direction sensitive Optical Module to be employed in deep sea neutrino telescopes

The NEMO (NEutrino Mediterranean Observatory, [I. Amore, [NEMO Collaboration], et al., Int. J. Mod. Phys. A 22 (2007) 3509, arXiv:0709.3991 [astro-ph]]) project studies, within the KM3NeT framework [U.F. Katz, Nucl. Instr. and Meth. A 567 (2006) 457, arXiv:astro-ph/0606068], new technologies for a km 3 -scale neutrino telescope in the Mediterranean Sea. The telescope goal is the investigation of the high energy component of the cosmic neutrino spectrum. The detection of these neutrinos is a promising tool for a better understanding of the mechanisms that originate the extreme energy cosmic rays. Neutrino energy and direction are reconstructed collecting the Cherenkov light produced in water by the muon coming from a neutrino interaction. Two prototypes of a new large area (10 in.) 4-anode photomultiplier, and manufactured by Hamamatsu, on request of the NEMO Collaboration, will be used for the first time to detect the direction of the detected Cherenkov light at the NEMO Capo Passero site.

Distributed compression and multiparty squashed entanglement

We study a protocol in which many parties use quantum communication to transfer a shared state to a receiver without communicating with each other. This protocol is a multiparty version of the fully quantum Slepian–Wolf protocol for two senders and arises through the repeated application of the two-sender protocol. We describe bounds on the achievable rate region for the distributed compression problem. The inner bound arises by expressing the achievable rate region for our protocol in terms of its vertices and extreme rays and, equivalently, in terms of facet inequalities. We also prove an outer bound on all possible rates for distributed compression based on the multiparty squashed entanglement, a measure of multiparty entanglement.

Projection onto polyhedra in outer representation

The projection of the origin onto an n-dimensional polyhedron defined by a system of m inequalities is reduced to a sequence of projection problems onto a one-parameter family of shifts of a polyhedron with at most m + 1 vertices in n + 1 dimensions. The problem under study is transformed into the projection onto a convex polyhedral cone with m extreme rays, which considerably simplifies the solution to an equivalent problem and reduces it to a single projection operation. Numerical results obtained for random polyhedra of high dimensions are presented.

Generating All Vertices of a 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. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. 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. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.

A validation and verification tool for global optimization solvers

Linearly constrained optimization models based on a systems description often possess multiple local optima. Therefore, there are a broad variety of problems in which the property of unique solution cannot be simply postulate or verified. The aim of global optimization (GO) is to find the best possible solution of multi-extremal problems. This paper illustrates the applicability of GO modeling techniques and solution strategies to real-world problems. In this paper we propose an enumeration approach for solving the linearly constrained optimization problem with continuous but general objective function. The method uses a parametric but unconstrained representation of the problem in terms of the vertices (and the extreme rays) of the feasible region. The unconstrained problem is then solved by the classical unconstrained continuous optimization procedure. Our aim is to propose a new introduction to optimization, the design of a general solution algorithm that is easy for the user to understand and provides useful information such as global bounding of the objective function. For illustrative and comparative purposes, the algorithm and its applications are presented in the context of applied problems solved by other solution algorithms.

Extreme lower probabilities

We consider lower probabilities on finite possibility spaces as models for the uncertainty about the state. These generalizations of classical probabilities can have some interesting properties; for example: k -monotonicity, avoiding sure loss, coherence, permutation invariance. The sets formed by all the lower probabilities satisfying zero or more of these properties are convex. We show how the extreme points and rays of these sets—the extreme lower probabilities—can be calculated and we give an illustration of our results.

Positive maps of second-order cones

Let and let be the n-dimensional second-order cone, or Lorentz cone. A linear map M from to is called positive if M[Lm]⊂Ln . The set of positive maps forms a convex cone, the positive cone. Its dual cone, the separable cone, is the convex hull of tensor products x⊗y, where x∈Ln , y∈Lm . The structure of the positive and the separable cones is the subject of the present contribution. We study the action of the automorphism group of the positive cone and provide a complete classification of its orbits by constructing explicitly canonical forms. This enables us to identify and to classify the extreme rays of the positive cone, as well as their dual counterparts, the largest faces of the separable cone. We also characterize positive maps which are scalable to doubly stochastic ones.

Benders, metric and cutset inequalities for multicommodity capacitated network design

Solving multicommodity capacitated network design problems is a hard task that requires the use of several strategies like relaxing some constraints and strengthening the model with valid inequalities. In this paper, we compare three sets of inequalities that have been widely used in this context: Benders, metric and cutset inequalities. We show that Benders inequalities associated to extreme rays are metric inequalities. We also show how to strengthen Benders inequalities associated to non-extreme rays to obtain metric inequalities. We show that cutset inequalities are Benders inequalities, but not necessarily metric inequalities. We give a necessary and sufficient condition for a cutset inequality to be a metric inequality. Computational experiments show the effectiveness of strengthening Benders and cutset inequalities to obtain metric inequalities.

Generating All Vertices of a 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. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. 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. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.

Concurrence revisited

Concurrence is a widely used entanglement measure of bipartite mixed states. We propose to consider the concurrence as being defined by a quadratic form on the space of self-adjoint operators. The square root of this form determines the values of the concurrence on the pure states, while the values on the mixed states are obtained as the largest convex extension, the convex roof. This viewpoint admits a generalization. Namely, the space of self-adjoint operators and the convex cone of positive semidefinite operators contained therein can be replaced by an arbitrary real vector space containing a convex cone. Then the concurrence is determined on the extreme rays of the cone by the square root of a quadratic form, and on the rest of the cone by the convex roof. We compute this generalized concurrence in the case when the cone is a second order cone. This enables us to compute the concurrence of arbitrary bipartite mixed states of rank 2. As an application, we compute the concurrences of the density matrices of all graphs with two edges or with three edges forming a triangle. We also consider the problem of maximizing the concurrence on the set of mixed states having a fixed spectrum.

The mixing-MIR set with divisible capacities

We study the set $$S = \{(x, y) \in \Re_{+} \times Z^{n} : x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}$$, where $$B_{j}, b_{j} \in \Re_{+} - \{0\}$$, j = 1, ..., n, and B 1 | ... | B n . The set S generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Gunluk and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about S remain unanswered, including the tractability of optimization over S and how to efficiently find a most violated cutting plane valid for P = conv (S). We address these questions by analyzing the extreme points and extreme rays of P. We give all extreme points and extreme rays of P. In the worst case, the number of extreme points grows exponentially with n. However, we show that, in some interesting cases, it is bounded by a polynomial of n. In such cases, it is possible to derive strong cutting planes for P efficiently. Finally, we use our results on the extreme points of P to give a polynomial-time algorithm for solving optimization over S.

Extreme rays in entangled edge states with positive partial transposes

Abstract In this Letter, we show that the 3 ⊗ 3 edge PPTES of the type ( 5 , 5 ) found by Clarisse and Ha generate extreme rays in the cone of all positive semi-definite matrices with positive semi-definite partial transposes.

Cones of closed alternating walks and trails

Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the alternating cone. The integral (respectively, {0, 1}) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices. The latter problem, called alternating reachability, is solved in a companion paper along with related results.

Implementing Generating Set Search Methods for Linearly Constrained Minimization

We discuss an implementation of a derivative-free generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints. The convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate. In the hard case, the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin, a difficult problem. We discuss here how state-of-the-art computational geometry methods make it tractable to solve this problem in connection with generating set search. We also discuss a number of other practical issues of implementation, such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set. We illustrate the behavior of the implementation on several problems from the CUTEr test suite. We have found it to be successful on problems with several hundred variables and linear constraints.