Maximum matching and a polyhedron with 0,1-vertices

Edmonds, Jack

doi:10.6028/jres.069b.013

Cited by 1,452 publications

(959 citation statements)

References 3 publications

Supporting

Mentioning

915

Contrasting

Unclassified

Order By: Relevance

“…The third one is a b-cut procedure based on blossom inequalities arising from Edmonds' description of matching-polytopes [5]. For this we just recall that the incidence vectors of u-capacitated b-matchings are the solutions of the following constraints:…”

Section: Linear Programming Formulationmentioning

confidence: 99%

Modeling the Geometry of the Endoplasmic Reticulum Network

Lemarchand

Euler

Lin

et al. 2014

Algorithms for Computational Biology

View full text Add to dashboard Cite

Abstract. We have studied the network geometry of the endoplasmic reticulum by means of graph theoretical and integer programming models. The purpose is to represent this structure as close as possible by a class of finite, undirected and connected graphs the nodes of which have to be either of degree three or at most of degree three. We determine plane graphs of minimal total edge length satisfying degree and angle constraints, and we show that the optimal graphs are close to the ER network geometry. Basically, two procedures are formulated to solve the optimization problem: a binary linear program, that iteratively constructs an optimal solution, and a linear program, that iteratively exploits additional cutting planes from different families to accelerate the solution process. All formulations have been implemented and tested on a series of real-life and randomly generated cases. The cutting plane approach turns out to be particularly efficient for the real-life testcases, since it outperforms the pure integer programming approach by a factor of at least 10.Keywords: endoplasmic reticulum, plane graph, 0-1 programming, separation procedure ProblemThe endoplasmic reticulum (ER) is a membrane-bound organelle that forms a highly complicated interconnected network of tubules and flattened sacs (known as cisternae) [10,6]. As the cortical ER in plant cells occupies a very thin, almost two-dimensional, layer of cytoplasm beneath the plasma membrane, our study of the ER network will be based on 2D approximations of the ER network in Tobacco leaf epidermal cells. Figure 1 (a) shows an instance of live cell images of an ER network [13]. Transition between tubules and cisternae can be highly dynamic and tubules can also dynamically change their polygonal network [10]. The dynamic ER shape is suggested to be adaptable to the cells requirements for ER function; for example, ER cisternae may be the preferred site of protein translocation while tubules might be the preferred site for ER vesicle budding. As an expanding number of proteins have been identified that mediate the generation and shape of the ER network, the stage is now set for investigations into the mechanisms regulating ER morphology within the cell. To be able to carry out such investigations, better tools are required to quantify the morphology and dynamic rearrangements that the ER undergoes. It is important to consider that whilst the proteins and stresses on the system driving formation and changes in the remodelling may differ between eukaryotic systems, the network geometry of the ER appears to be fairly well conserved in terms of it consisting of a polygonal network of interconnected tubules and cisternae. Therefore, the problem considered here has universal appeal towards understanding the constraints placed on ER network formation in all systems.A quantitative analysis [11] of the ER network in tobacco leaf epidermal cells suggests that it is a perturbed Euclidean Steiner network between terminals, where terminals are persistent nodes (static elemen...

show abstract

Section: Linear Programming Formulationmentioning

confidence: 99%

Modeling the Geometry of the Endoplasmic Reticulum Network

Lemarchand

Euler

Lin

et al. 2014

Algorithms for Computational Biology

View full text Add to dashboard Cite

show abstract

“…Pk where the first inequality follows from the complete description of the perfect matching polytope due to Edmonds [8] and the first equality follows from the parsimonious property. Combining equations (9), (10) and (11) This corollary also generalizes the result on the worst-case analysis of the Steiner tree problem.…”

Section: Fork=l Top Domentioning

confidence: 99%

Survivable networks, linear programming relaxations and the parsimonious property

Goemans

Bertsimas

1993

Mathematical Programming

151

View full text Add to dashboard Cite

We consider the survivable network design problem --the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-edge-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worst-case analyses of two heuristics for the survivable network design problem.

show abstract

“…We use minimum-weight perfect matching [22][23][24] to decode to physical errors, based on the pattern of detection events and an error model for the system. Intuitively, it connects detection events in pairs or to the boundary using the shortest weighted path length.…”

mentioning

confidence: 99%

State preservation by repetitive error detection in a superconducting quantum circuit

Kelly

Barends

Fowler

et al. 2015

Nature

886

1,004

View full text Add to dashboard Cite

Quantum computing becomes viable when a quantum state can be preserved from environmentally-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) 1-6 is capable of identifying and correcting them. Adding more qubits improves the preservation by guaranteeing increasingly larger clusters of errors will not cause logical failure -a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here, we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural precursor of the twodimensional surface code QEC scheme 7 , and track errors as they occur by repeatedly performing projective quantum non-demolition (QND) parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 for five qubits and a factor of 8.5 for nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger-Horne-Zeilinger (GHZ) state. The successful suppression of environmentally-induced errors strongly motivates further research into the many exciting challenges associated with building a large-scale superconducting quantum computer.The ability to withstand multiple errors during computation is a critical aspect of error correction. We define n-th order fault-tolerance to mean that any combination of n errors is tolerable. Previous experiments based on nuclear magnetic resonance 8,9 , ion traps 10 , and superconducting circuits [11][12][13] have demonstrated multi-qubit states that are first-order tolerant to one type of error. Recently, experiments with ion traps and superconducting circuits have shown the simultaneous detection of multiple types of errors 14,15 . The above hallmark experiments demonstrate error correction in a single round; however, quantum information must be preserved throughout computation using multiple error correction cycles. The basics of repeating cycles have been shown in ion traps 16 and superconducting circuits 17 . Until now, it has been an open challenge to combine these elements to make the information stored in a quantum system robust against errors which intrinsically arise from the environment.The key to detecting errors in quantum information is to perform QND parity measurements. In the surface code, this is done by arranging qubits in a chequerboard pattern -with data qubits corresponding to the white, and measure qubits to the black squares (see Fig. 1) -and using these ancilla measure qubits to repetitively perform parity measurements to detect bit-flip (X) and phase-flip (Ẑ) errors 7 . A square chequerboard with (4n + 1) 2 qubits is n-th order fault tolerant, meaning at least n+1 errors must occur to cause failure in preserving a state if fidelities are above a threshold. W...

show abstract

Maximum matching and a polyhedron with 0,1-vertices

Cited by 1,452 publications

References 3 publications

Modeling the Geometry of the Endoplasmic Reticulum Network

Modeling the Geometry of the Endoplasmic Reticulum Network

Survivable networks, linear programming relaxations and the parsimonious property

State preservation by repetitive error detection in a superconducting quantum circuit

Contact Info

Product

Resources

About