Short Presburger Arithmetic Is Hard

Nguyen, Danny; Pak, Igor

doi:10.1109/focs.2017.13

Cited by 9 publications

(12 citation statements)

References 33 publications

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“…In , certain subclasses of classical PA formulas, called short PA formulas , were investigated. The PA formulas in each such subclass are allowed to have only a bounded number of variables, quantifiers and inequalities (atomic formulas).…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…Also without affecting the complexity, we can assume that

g_{1} = ν, h_{1} = 1, e_{1} = 0

, i.e.,

{AP}_{1} = false{ν false}

. The main argument in uses continued fractions to construct an integer M and a rational number

p / q

such that the best approximations of

p / q

, in the terminology of continued fractions, encode

⋃_{i = 1}^{n} {AP}_{i}

modulo M . The main point is that

p / q

should satisfy

⌊ p / q ⌋ = g_{1} = ν

, so that

[μ, ν] = [1, p / q]

, and the formula

\begin{matrix} left Φ_{p, q, M} (z) = 1 \leq z \leq p / q \land \exists boldy y_{2} \equiv z (\mod M) & left \land ⌊ p / q ⌋ \leq y_{2} < p \land q y_{2} < p y_{1} \land \\ left \forall boldx \neg ({}, \begin{matrix} p y_{1} - q y_{2} \geq p x_{1} - q x_{2} \geq 0 \\ y_{2} > x_{2} > 0 \end{matrix}) \end{matrix}

satisfies the property

…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…Combined with the positive results in , this settled the last open subcase of classical PA complexity problems. The main reduction in started with the following

boldNP

‐complete problem:

\begin{matrix} true \\ left Given an {interval}^{2} [μ, ν] \subset double-struckZ and n arithmetic progressions {AP}_{i} = \\ left AP (g_{i}, h_{i}, e_{i}) : = {g_{i}, g_{i} + e_{i}, \dots, g_{i} + h_{i} e_{i}}, with 1 \leq μ \leq ν, g_{i}, h_{i}, e_{i} \in double-struckZ, \\ left h_{i} \geq 1, decide if there exists some z \in false[μ, ν false] ∖ ⋃_{i = 1}^{n} {AP}_{i} . \end{matrix}

…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…We shall present what amount to two different proofs of Theorem in the following two sections. In each case, we leverage the main result of Nguyen and Pak which yields a 3‐parametric Σ 2 PA formula, and then show how this can be reduced to a 2‐parametric Σ 2 PA formula whose points are equally “hard” to count (modulo polynomial‐time reductions). The first reduction we present, in § , uses a trick due to Glivický and Pudlák to encode multiplication by three different integers using multiplication by only two integers, and this reduction has the advantage of not increasing the number of free variables in the formula.…”

Section: Introductionmentioning

confidence: 99%

“…The curve

scriptC

includes

(p, q)

in , but not here. This small difference is not very significant as one can easily check.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Bogart

Goodrick

Nguyen

et al. 2019

Mathematical Logic Qtrly

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We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set Sboldt⊆double-struckZd as boldt varies in Zk, and we examine the counting function false|Sboldtfalse| as a function of t. For a single parameter, it is known that false|Stfalse| can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming P≠NP) we construct a parametric set St1,t2 such that false|St1,t2false| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets Sboldt⊆double-struckZd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that false|Sboldtfalse| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.

show abstract

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…Also without affecting the complexity, we can assume that

g_{1} = ν, h_{1} = 1, e_{1} = 0

, i.e.,

{AP}_{1} = false{ν false}

. The main argument in uses continued fractions to construct an integer M and a rational number

p / q

such that the best approximations of

p / q

, in the terminology of continued fractions, encode

⋃_{i = 1}^{n} {AP}_{i}

modulo M . The main point is that

p / q

should satisfy

⌊ p / q ⌋ = g_{1} = ν

, so that

[μ, ν] = [1, p / q]

, and the formula

\begin{matrix} left Φ_{p, q, M} (z) = 1 \leq z \leq p / q \land \exists boldy y_{2} \equiv z (\mod M) & left \land ⌊ p / q ⌋ \leq y_{2} < p \land q y_{2} < p y_{1} \land \\ left \forall boldx \neg ({}, \begin{matrix} p y_{1} - q y_{2} \geq p x_{1} - q x_{2} \geq 0 \\ y_{2} > x_{2} > 0 \end{matrix}) \end{matrix}

satisfies the property

…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

“…Combined with the positive results in , this settled the last open subcase of classical PA complexity problems. The main reduction in started with the following

boldNP

‐complete problem:

\begin{matrix} true \\ left Given an {interval}^{2} [μ, ν] \subset double-struckZ and n arithmetic progressions {AP}_{i} = \\ left AP (g_{i}, h_{i}, e_{i}) : = {g_{i}, g_{i} + e_{i}, \dots, g_{i} + h_{i} e_{i}}, with 1 \leq μ \leq ν, g_{i}, h_{i}, e_{i} \in double-struckZ, \\ left h_{i} \geq 1, decide if there exists some z \in false[μ, ν false] ∖ ⋃_{i = 1}^{n} {AP}_{i} . \end{matrix}

…”

Section: Proof Of Theorem and Its Corollariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The curve

scriptC

includes

(p, q)

in , but not here. This small difference is not very significant as one can easily check.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Bogart

Goodrick

Nguyen

et al. 2019

Mathematical Logic Qtrly

Self Cite

View full text Add to dashboard Cite

show abstract

On the Number of Integer Points in Translated and Expanded Polyhedra

Nguyen¹,

Pak²

2020

Discrete Comput Geom

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We prove that the problem of minimizing the number of integer points in parallel translations of a rational convex polytope in R 6 is NP-hard. We apply this result to show that given a rational convex polytope P ⊂ R 6 , finding the largest integer t s.t. the expansion tP contains fewer than k integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations. Integer Point Minimization (IPM) Input:AParametric polytopes P b := {x ∈ R n : Ax ≤ b} were introduced by Kannan [Kan90], who gave a polynomial time algorithm for IPM with k = 0 and n bounded. For larger fixed values k, Aliev, De Loera and Louveaux [ADL16] proved that IPM is also polynomial time by employing the short generating functions technique by Barvinok and Woods [BW03] (see also [Bar06b,Bar08]). The following problem is an especially attractive special case:

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Sparse representation of vectors in lattices and semigroups

et al. 2021

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We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

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Short Presburger Arithmetic Is Hard

Cited by 9 publications

References 33 publications

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

On the Number of Integer Points in Translated and Expanded Polyhedra

Sparse representation of vectors in lattices and semigroups

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