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

Abstract

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...