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The complexity of disjunctive linear Diophantine constraints
[article]

2018
*
arXiv
*
pre-print

We study

arXiv:1807.00985v1
fatcat:aalxc72wffbdvktsnic2abreme
*the**Constraint*Satisfaction Problem CSP(A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete. ...*The**complexity**of**disjunctive**linear**Diophantine**constraints*◮ Proposition 4. Let A be a finite-signature first-order reduct*of*(Z; +, 1) that contains + and is a core. ... Now let us address*the**complexity**of**the*algorithm. ...##
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The Complexity of Disjunctive Linear Diophantine Constraints

2018
*
International Symposium on Mathematical Foundations of Computer Science
*

We study

doi:10.4230/lipics.mfcs.2018.33
dblp:conf/mfcs/BodirskyMMM18
fatcat:r2awaatt2feytfzfbtzbwcj7em
*the**Constraint*Satisfaction Problem CSP(A), where A is first-order definable in (Z; +, 1) and contains +. We prove such problems are either in P or NP-complete. ... S 2 0 1 8 33:4*The**Complexity**of**Disjunctive**Linear**Diophantine**Constraints*write λ • g for (λg 1 , . . . , λg k ). ... M F C S 2 0 1*Complexity**of**Disjunctive**Linear**Diophantine**Constraints*Proof.*The*two items are clear from*the*proof*of*Lemma 5. ...##
###
New Complexity Results for Some Linear Counting Problems Using Minimal Solutions to Linear Diophantine Equations
[chapter]

2003
*
Lecture Notes in Computer Science
*

Using results on minimal solutions (in nonnegative integers) for

doi:10.1007/3-540-45089-0_16
fatcat:4ezy6onw7zhl7mhcqo2fltrg7a
*linear**Diophantine*systems, we obtain new*complexity*results for*the*problem, as well as for other*linear*counting problems*of*finite state ... In contrast to previously known results,*the**complexity*bounds obtained in this paper are polynomial in*the*size*of**the*transition system in consideration, when*the**linear**constraint*is fixed. ... From this instance and*the*given , we will construct an ordered finite*linear**Diophantine*equation system (i.e.,*the**disjunctive*representations*of**the*coefficient matrix for system (12) and c to ...##
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Knuth-Bendix constraint solving is NP-complete
[article]

2002
*
arXiv
*
pre-print

We show

arXiv:cs/0207068v1
fatcat:3nkk43lfzjfnxi4oewbwai2foi
*the*NP-completeness*of**the*existential theory*of*term algebras with*the*Knuth-Bendix order by giving a nondeterministic polynomial-time algorithm for solving Knuth-Bendix ordering*constraints*. ... Then in Section 4 we show how to transform any*constraint*in isolated form into an equivalent*disjunction**of*systems*of**linear**Diophantine*inequalities on*the*weights*of*variables. ... We will show how these statements can be expressed using systems*of**linear**Diophantine*inequalities on*the*weights*of*variables and then use*the*fact that*the*decidability*of*systems*of**linear**Diophantine*...##
###
Linear reachability problems and minimal solutions to linear Diophantine equation systems

2004
*
Theoretical Computer Science
*

Using some known results on minimal solutions (in nonnegative integers) for

doi:10.1016/j.tcs.2004.07.015
fatcat:3qyzraiizjfzhd4trrrufuvq6m
*linear**Diophantine*equation systems, we present new time*complexity*bounds for*the*problem. ... In contrast to*the*previously known results,*the*bounds obtained in this paper are polynomial in*the*size*of**the*transition system in consideration, when*the**linear**constraint*is fixed. ... Acknowledgements*The*authors would like to thank P. San Pietro and*the*anonymous referees for many valuable suggestions. ...##
###
From Elliott-MacMahon to an algorithm for general linear constraints on naturals
[chapter]

1995
*
Lecture Notes in Computer Science
*

We show how this redundancy may be eliminated at

doi:10.1007/3-540-60299-2_2
fatcat:rtjrcfbq7bd3xpmyfxsw2pzy7i
*the*cost*of*an increase in*the*space*complexity*. keywords: integer programming,*linear**diophantine**constraints*? ... We describe a new algorithm for solving a conjunction*of**linear**diophantine*equations, inequations and disequations in natural numbers. ... A system*of**linear**diophantine**constraints*is a nite conjunction*of**constraints*. ...##
###
Approaching Arithmetic Theories with Finite-State Automata
[chapter]

2020
*
Lecture Notes in Computer Science
*

This approach has recently been instrumental for settling long-standing open problems about

doi:10.1007/978-3-030-40608-0_3
fatcat:y4yvocbpvzdcba4jlsttgv2nwq
*the**complexity**of*deciding*the*existential fragments*of*Büchi arithmetic and*linear*arithmetic over p-adic fields ...*The*automata-theoretic approach provides an elegant method for deciding*linear*arithmetic theories. ... It is well-known that*the*set*of*nonnegative integer solutions*of*a system*of**linear**Diophantine*equations is a semi-*linear*set [7] . ...##
###
Page 738 of Mathematical Reviews Vol. 34, Issue 3
[page]

1967
*
Mathematical Reviews
*

Assuming that all prime implicants have

*the*same cost,*the*author gives a new method, involving use*of*minimal*constraint**linear**diophantine*programming, for*the*selec- tion*of*a minimal cost subset*of*... This note continues*the*author’s paper on*the*comparison*of**the**complexities**of*“‘tupikovye’’ (previously translated 738 INFORMATION AND COMMUNICATION, SYSTEMS as ‘“minimal-essential’’) and minimal*disjunctive*...##
###
Unification of infinite sets of terms schematized by primal grammars

1997
*
Theoretical Computer Science
*

Since schematizations provide a different approach to solve

doi:10.1016/s0304-3975(96)00052-7
fatcat:creronyxcnhthbybl5g6ktzlm4
*the*same kind*of*problems as*constraints*do, they can be viewed as a new type*of**constraints*. ...*The*paper presents a new recurrent schematization called primal grammars.*The*main idea behind*the*primal grammars is to use primitive recursion as*the*generating engine*of*infinite sets. ... Kolaitis and Pierre Lescanne for their comments on an earlier draft*of*this paper. We are also indebted to an anonymous referee that discovered an error in a previous version*of*Theorem 5.12. ...##
###
Diophantine equations, Presburger arithmetic and finite automata
[chapter]

1996
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/3-540-61064-2_27
fatcat:7xyyf43dwrbgdf4p5js5wt2pbi
*the*use*of*nite automata provides a decision procedure for Presburger Arithmetic with optimal worst case*complexity*. ...*The*special case*of**linear**Diophantine*equations has been studied even earlier 2]. ... Let e be*the**linear**Diophantine*equation a 1 x 1 + +a n x n = k. ...##
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On the Complexity of Selecting Disjunctions in Integer Programming

2010
*
SIAM Journal on Optimization
*

mixed integer

doi:10.1137/080737587
fatcat:rwwwbtauabcinmavk4k3utbrta
*linear*programs. ... We first consider*the*problem*of*selecting a general*disjunction*and show that*the*problem*of*selecting an optimal such*disjunction*, according to specific criteria described herein, is N P-hard. ... Acknowledgements:*The*authors would like to acknowledge*the*anonymous referees whose suggestions led to a considerable improvement in*the*presentation*of**the*paper. ...##
###
On the Complexity of Branching Proofs

2020
*
Computational Complexity Conference
*

Our recompilation techniques works by first replacing each

doi:10.4230/lipics.ccc.2020.34
dblp:conf/coco/DadushT20
fatcat:pgnwxvkbtrbxnpuvmoamfwzv4y
*disjunction*using an iterated*Diophantine*approximation, introduced by Frank and Tardos (Combinatorica 1986), and proceeds by "fixing up"*the*... We resolve this question in*the*affirmative, by showing that any branching proof can be recompiled so that*the*normals*of**the**disjunctions*have coefficients*of*size at most (n R)^O(n²), where R ∈ ℕ is ... Here, T includes*the*bit-size*of*all*the**disjunctions*ax ≤ b or ≥ b + 1 used 34:4 On*the**Complexity**of*Branching Proofs in T . ...##
###
On the Complexity of Branching Proofs
[article]

2020
*
arXiv
*
pre-print

In a general branching proof, one constructs a branching tree by adding an integer

arXiv:2006.04124v1
fatcat:woceg6ycgnerlaj735fgjgewc4
*disjunction*ax≤ b or ax≥ b+1, a∈Z^n, b ∈Z, at each node, such that*the*leaves*of**the*tree correspond to empty sets (i.e ... We resolve this question by showing that any branching proof can be recompiled so that*the*integer*disjunctions*have coefficients*of*size at most (n R)^O(n^2), where R ∈N such that K ∈ R B_1^n, while increasing ... However, for a general branching proof T ,*the*tree size |T | ignores*the*"*complexity*"*of**the*individual*disjunctions*. ...##
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Group equations with abelian predicates
[article]

2022
*
arXiv
*
pre-print

We extend

arXiv:2204.13946v1
fatcat:h4sqytpgpvhklbtlq3npa6clb4
*the*line*of*work on word equations with length*constraints*, and more generally, on extensions*of**the*existential theory*of*semigroups, to*the*world*of*groups. ... In this paper we begin*the*systematic study*of*group equations with abelian predicates in*the*main classes*of*groups where solving equations is possible. ...*The*second named author was supported by*the*ERC grant PCG-336983, by*the*Basque Government grant IT974-16, by*the*Ministry*of*Economy, Industry and Competitiveness*of**the*Spanish Government Grant MTM2017 ...##
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A Solvable Class of Quadratic Diophantine Equations with Applications to Verification of Infinite-State Systems
[chapter]

2003
*
Lecture Notes in Computer Science
*

A¨-system consists

doi:10.1007/3-540-45061-0_53
fatcat:npwl7e5ik5h6fpitbgdvk3teci
*of*¨quadratic*Diophantine*equations over nonnegative integer variables ...*The*authors would like to thank*the*anonymous referees for many valuable comments and suggestions. ... It is known [12] that there is a fixed 1 such that there is no algorithm to solve*Diophantine*systems in*the*following form: (4) is a*disjunction**of*formulas in*the*form*of*(6), is decidable. ¦ # ...
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