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The complexity of disjunctive linear Diophantine constraints [article]

Manuel Bodirsky, Barnaby Martin, Marcello Mamino, Antoine Mottet
2018 arXiv   pre-print
We study 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.  ... 
arXiv:1807.00985v1 fatcat:aalxc72wffbdvktsnic2abreme

The Complexity of Disjunctive Linear Diophantine Constraints

Manuel Bodirsky, Barnaby Martin, Marcello Mamino, Antoine Mottet, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
We study 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.  ... 
doi:10.4230/lipics.mfcs.2018.33 dblp:conf/mfcs/BodirskyMMM18 fatcat:r2awaatt2feytfzfbtzbwcj7em

New Complexity Results for Some Linear Counting Problems Using Minimal Solutions to Linear Diophantine Equations [chapter]

Gaoyan Xie, Cheng Li, Zhe Dang
2003 Lecture Notes in Computer Science  
Using results on minimal solutions (in nonnegative integers) for 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  ... 
doi:10.1007/3-540-45089-0_16 fatcat:4ezy6onw7zhl7mhcqo2fltrg7a

Knuth-Bendix constraint solving is NP-complete [article]

Konstantin Korovin, Andrei Voronkov
2002 arXiv   pre-print
We show 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  ... 
arXiv:cs/0207068v1 fatcat:3nkk43lfzjfnxi4oewbwai2foi

Linear reachability problems and minimal solutions to linear Diophantine equation systems

Gaoyan Xie, Cheng Li, Zhe Dang
2004 Theoretical Computer Science  
Using some known results on minimal solutions (in nonnegative integers) for 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.  ... 
doi:10.1016/j.tcs.2004.07.015 fatcat:3qyzraiizjfzhd4trrrufuvq6m

From Elliott-MacMahon to an algorithm for general linear constraints on naturals [chapter]

Eric Domenjoud, Ana Paula Tomás
1995 Lecture Notes in Computer Science  
We show how this redundancy may be eliminated at 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.  ... 
doi:10.1007/3-540-60299-2_2 fatcat:rtjrcfbq7bd3xpmyfxsw2pzy7i

Approaching Arithmetic Theories with Finite-State Automata [chapter]

Christoph Haase
2020 Lecture Notes in Computer Science  
This approach has recently been instrumental for settling long-standing open problems about 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] .  ... 
doi:10.1007/978-3-030-40608-0_3 fatcat:y4yvocbpvzdcba4jlsttgv2nwq

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

Miki Hermann, Roman Galbavý
1997 Theoretical Computer Science  
Since schematizations provide a different approach to solve 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.  ... 
doi:10.1016/s0304-3975(96)00052-7 fatcat:creronyxcnhthbybl5g6ktzlm4

Diophantine equations, Presburger arithmetic and finite automata [chapter]

Alexandre Boudet, Hubert Comon
1996 Lecture Notes in Computer Science  
We show that 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.  ... 
doi:10.1007/3-540-61064-2_27 fatcat:7xyyf43dwrbgdf4p5js5wt2pbi

On the Complexity of Selecting Disjunctions in Integer Programming

Ashutosh Mahajan, Ted Ralphs
2010 SIAM Journal on Optimization  
mixed integer 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.  ... 
doi:10.1137/080737587 fatcat:rwwwbtauabcinmavk4k3utbrta

On the Complexity of Branching Proofs

Daniel Dadush, Samarth Tiwari, Shubhangi Saraf
2020 Computational Complexity Conference  
Our recompilation techniques works by first replacing each 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 .  ... 
doi:10.4230/lipics.ccc.2020.34 dblp:conf/coco/DadushT20 fatcat:pgnwxvkbtrbxnpuvmoamfwzv4y

On the Complexity of Branching Proofs [article]

Daniel Dadush, Samarth Tiwari
2020 arXiv   pre-print
In a general branching proof, one constructs a branching tree by adding an integer 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.  ... 
arXiv:2006.04124v1 fatcat:woceg6ycgnerlaj735fgjgewc4

Group equations with abelian predicates [article]

Laura Ciobanu, Albert Garreta
2022 arXiv   pre-print
We extend 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  ... 
arXiv:2204.13946v1 fatcat:h4sqytpgpvhklbtlq3npa6clb4

A Solvable Class of Quadratic Diophantine Equations with Applications to Verification of Infinite-State Systems [chapter]

Gaoyan Xie, Zhe Dang, Oscar H. Ibarra
2003 Lecture Notes in Computer Science  
A¨-system consists 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. ¦ #  ... 
doi:10.1007/3-540-45061-0_53 fatcat:npwl7e5ik5h6fpitbgdvk3teci
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