Parallel Parametric Linear Programming Solving, and Application to Polyhedral Computations

Coti, Camille; Monniaux, David; Yu, Hang

doi:10.1007/978-3-030-22750-0_52

Cited by 4 publications

(6 citation statements)

References 20 publications

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“…Then, for given values of the

μ_{i}

, the problem is either unbounded, or there is one single optimal solution X ∗ . It can be shown that the region of the

(μ_{1}, \dots, μ_{k})

associated to a given optimum X ∗ is a convex polyhedron (for C 0 , a convex polyhedral cone), and that these regions form a quasi‐partition of the space of parameters (two regions may overlap at their boundary, but not in their interior) 14,19,27 . The output of the parametric linear programming solver is this quasi‐partition, and the associated optima—in our applications, the problem is always bounded in the optimization directions, so we do not deal with the unbounded case.…”

Section: Sequential Algorithmsmentioning

confidence: 99%

“…We are currently investigating approaches for getting rid of degeneracy—enforcing one optimal vector X ∗ and only one optimal basis per vector D , except at the boundaries. The methods for doing so rely on lexicographic orderings or perturbations on the objective and/or constant term, or pivoting rules 14 . We have recently proposed a working solution to degeneracy 28 but there is still room for improvement.…”

Section: Sequential Algorithmsmentioning

confidence: 99%

“…In addition to computations on convex polyhedra, parametric linear programming is also used for control applications: 13,14 instead of using a solver, whose computation times are high and hard to predict, inside the control loop, the solution of the linear program is tabulated as a piecewise linear function over the values of the parameters. Parametric linear programming is also used for affine linear approximations of nonlinear expressions 3 …”

Section: Introductionmentioning

confidence: 99%

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A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

Coti

Monniaux

2020

Concurrency and Computation

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Parametric linear programming is a central operation for polyhedral computations, as well as in certain control applications. Here, we propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems. This type of parallel applications is challenging, because several tasks might be computing the same region. In this article, we are presenting the algorithm itself with a parallel redundancy elimination algorithm, and conducting a thorough performance analysis. K E Y W O R D S parallel parametric linear programming, polyhedra 1 INTRODUCTION A convex polyhedron in dimension n is the solution set over Q n (or, equivalently, 1 R n) of a system of inequalities (with integer or rational coefficients). Since all the polyhedra we consider here are convex, we shall talk of polyhedra for short. There also exist integer polyhedra, defined over Z n , but they are very different in many respects; we shall not consider them here. 2 Computations over polyhedra arise in low dimension (n = 2 and n = 3) for modeling physical objects, but here we are interested in higher dimensions. Polyhedra in higher dimension are typically used to enclose the set of reachable states of systems whose state can be expressed, at least partially, as a vector of reals or rationals. For instance, one can study the flow of ordinary differential equations, or more generally the trajectories of hybrid systems, by enclosing the trajectories into a succession of convex polyhedra. If the internal state of a control system is expressed by a vector of n numeric variables, one may prove that this system never encounters a bad condition by exhibiting a polyhedron P such that the initial state belongs to P, no bad condition belongs to P, and all possible time steps of the control system leave P stable (i.e., it is not possible to execute a step starting in P and ending outside of P). One generally proves the correctness of programs by exhibiting inductive invariants-an inductive invariant for a loop is a set containing the precondition of the loop, stable by moving to the next loop iteration, and implying the desired postcondition. Since providing inductive invariants by hand is tedious, it is desirable to automate that process. One approach for doing so is abstract interpretation, where an ascending sequence of sets of states is computed until reaching a fixed point. One may, for instance, search such sets as products of intervals, an approach known as interval analysis, but the lack of relationships between the dimensions tends to severely limit the kind of properties that can be proved (e.g., one cannot have an invariant i < n where n is a parameter: this is neither an interval on i nor on n nor a combination thereof). Cousot and Halbwachs proposed searching for polyhedral inductive invariants. 1,2 The operations needed there are projection, more generally image by an affine transformation, convex hull, and inclusion (or equality) test, together with an extrapolation operator known as widening. 1 Whether emptiness tests, inclusio...

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“…Then, for given values of the

μ_{i}

, the problem is either unbounded, or there is one single optimal solution X ∗ . It can be shown that the region of the

(μ_{1}, \dots, μ_{k})

Section: Sequential Algorithmsmentioning

confidence: 99%

Section: Sequential Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

Coti

Monniaux

2020

Concurrency and Computation

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“…We extract the indices of the basic variables from that tableau; M B and O B denote the sub-matrices from M and B containing only the columns corresponding to the basic variables. By linear algebra in rational arithmetic 5 we compute a matrix Θ, representing the substitution performed by the simplex algorithm. Then we apply this substitution to the objective matrix O to get the new objective function…”

Section: Obtaining Rational Solutionmentioning

confidence: 99%

“…Furthermore, the solving is divided into independent tasks, which may be scheduled in parallel. The parallel implementation is covered in [5].…”

Section: Introduction and Related Workmentioning

confidence: 99%

An Efficient Parametric Linear Programming Solver and Application to Polyhedral Projection

Monniaux

2019

Lecture Notes in Computer Science

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Polyhedral projection is a main operation of the polyhedron abstract domain. It can be computed via parametric linear programming (PLP), which is more efficient than the classic Fourier-Motzkin elimination method.In prior work, PLP was done in arbitrary precision rational arithmetic. In this paper, we present an approach where most of the computation is performed in floating-point arithmetic, then exact rational results are reconstructed.We also propose a workaround for a difficulty that plagued previous attempts at using PLP for computations on polyhedra: in general the linear programming problems are degenerate, resulting in redundant computations and geometric descriptions.

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Parallel maximal common subgraphs with labels for molecular biology

Agbeto,

Coti,

Reinharz

2024

Preprint

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Advances in graph algorithmics have allowed in-depth study of many natural objects from molecular biology or chemistry to social networks. Particularly in molecular biology and cheminformatics, understanding complex structures by identifying conserved sub-structures is a key milestone towards the artificial design of novel components with specific functions. Given a dataset of structures, we are interested in identifying all maximum common connected partial subgraphs between each pair of graphs, a task notoriously NP-Hard. In this work, we present 3 parallel algorithms over shared and distributed memory to enumerate all maximal connected common sub-graphs between pairs of arbitrary multi-directed graphs with labels on their edges. We offer an implementation of these methods and evaluate their performance on the non-redundant dataset of all known RNA 3D structures. We show that we can compute the exact results in a reasonable time for each pairwise comparison while taking into account a much more diverse set of interactions---resulting in much denser graphs---resulting in an order of magnitude more conserved modules. All code is available at https://gitlab.info.uqam.ca/cbe/pasigraph and results in the branch results.

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Parallel Parametric Linear Programming Solving, and Application to Polyhedral Computations

Abstract: Parametric linear programming is central in polyhedral computations and in certain control applications. We propose a task-based scheme for parallelizing it, with quasi-linear speedup over large problems.

Cited by 4 publications

References 20 publications

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

A task‐based approach to parallel parametric linear programming solving, and application to polyhedral computations

An Efficient Parametric Linear Programming Solver and Application to Polyhedral Projection

Parallel maximal common subgraphs with labels for molecular biology

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