Computing the Integer Points of a Polyhedron, I: Algorithm

Jing, Rui-Juan; Maza, Marc Moreno

doi:10.1007/978-3-319-66320-3_17

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…is a finite union of sets of the form V + h where V is a linear subspace of Q n . When we intersect these translated subspaces with requirements that odd coordinates must be strictly positive and distinct, we get a set of linear (in)equalities, for which an integer solution x can be found using a variation of integer linear programming (see, e.g., [JMM17]). If ∥x∥ > b 0 , then set b 1 = ⌈∥x∥⌉.…”

Section: The Algorithmmentioning

confidence: 99%

What’s Decidable About Discrete Linear Dynamical Systems?

Karimov

Kelmendi

Ouaknine

et al. 2022

Lecture Notes in Computer Science

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We consider reachability decision problems for linear dynamical systems: Given a linear map on R d , together with source and target sets, determine whether there is a point in the source set whose orbit, obtained by repeatedly applying the linear map, enters the target set. When the source and target sets are semialgebraic, this problem can be reduced to a point-to-polytope reachability question. The latter is generally believed not to be substantially harder than the well-known Skolem and Positivity Problems. The situation is markedly different for multiple reachability, i.e. the question of whether the orbit visits the target set at least m times, for some given positive integer m. In this paper, we prove that when the source set is semialgebraic and the target set consists of a hyperplane, multiple reachability is undecidable; in fact we already obtain undecidability in ambient dimension d = 10 and with fixed m = 9. Moreover, as we observe that procedures for dimensions 3 up to 9 would imply strong results pertaining to effective solutions of Diophantine equations, we mainly focus on the affine plane (R 2 ). We obtain two main positive results. We show that multiple reachability is decidable for halfplane targets, and that it is also decidable for general semialgebraic targets, provided the linear map is a rotation. The latter result involves a new method, based on intersections of algebraic subgroups with subvarieties, due to Bombieri and Zannier.

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Section: The Algorithmmentioning

confidence: 99%

What’s Decidable About Discrete Linear Dynamical Systems?

Karimov

Kelmendi

Ouaknine

et al. 2022

Lecture Notes in Computer Science

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“…This representation is useful for finding solutions of linear inequality systems. The projected representation was introduced in [18,19] and will be reviewed in Definition 23.…”

Section: Minimal Representation Of the Projected Polyhedronmentioning

confidence: 99%

“…This initial redundancy test cone is used to remove all the redundant inequalities in the input polyhedron. Moreover, our method has a better algebraic complexity estimate than the approaches using linear programming; see [18,19] for estimates of those approaches.…”

Section: Introductionmentioning

confidence: 99%

Complexity Estimates for Fourier-Motzkin Elimination

Jing

Moreno-Maza

Talaashrafi

2018

Preprint

Self Cite

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In this paper, we propose a new method for removing all the redundant inequalities generated by Fourier-Motzkin elimination. This method is based on an improved version of Balas' work and can also be used to remove all the redundant inequalities in the input system. Moreover, our method only uses arithmetic operations on matrices and avoids resorting to linear programming techniques. Algebraic complexity estimates and experimental results show that our method outperforms alternative approaches, in particular those based on linear programming and simplex algorithm.

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The Z_Polyhedra Library in Maple

Jing

Maza

2020

Communications in Computer and Information Science

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Computing the Integer Points of a Polyhedron, I: Algorithm

Cited by 7 publications

References 14 publications

What’s Decidable About Discrete Linear Dynamical Systems?

What’s Decidable About Discrete Linear Dynamical Systems?

Complexity Estimates for Fourier-Motzkin Elimination

The Z_Polyhedra Library in Maple

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