Theory of linear and integer programming
Theory of linear and integer programming
An algorithm for linear programming which requires O(((m+n)n2+(m+n)1.5n)L) arithmetic operations
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Polyhedral functions and multiparametric linear programming
Journal of Optimization Theory and Applications
The complexity of linear problems in fields
Journal of Symbolic Computation
Real quantifier elimination is doubly exponential
Journal of Symbolic Computation
Solution of parametrized linear inequalities by fourier elimination and its applications
Journal of Optimization Theory and Applications
Partial Cylindrical Algebraic Decomposition for quantifier elimination
Journal of Symbolic Computation
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
REDLOG: computer algebra meets computer logic
ACM SIGSAM Bulletin
Parameterized polyhedra and their vertices
International Journal of Parallel Programming
A New Approach for Automatic Theorem Proving in Real Geometry
Journal of Automated Reasoning
An Analysis of Zero-Clairvoyant Scheduling
TACAS '02 Proceedings of the 8th International Conference on Tools and Algorithms for the Construction and Analysis of Systems
A new polynomial-time algorithm for linear programming
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
QEPCAD B: a program for computing with semi-algebraic sets using CADs
ACM SIGSAM Bulletin
An Analysis of Totally Clairvoyant Scheduling
Journal of Scheduling
Verification of safety properties for parameterized regular systems
ACM Transactions on Embedded Computing Systems (TECS)
Tractable Fragments of Presburger Arithmetic
Theory of Computing Systems
Efficient solving of quantified inequality constraints over the real numbers
ACM Transactions on Computational Logic (TOCL)
On a decision procedure for quantified linear programs
Annals of Mathematics and Artificial Intelligence
Relatively quantified constraint satisfaction
Constraints
A solver for QBFs in negation normal form
Constraints
Value ordering for quantified CSPs
Constraints
The complexity of quantified constraint satisfaction problems under structural restrictions
IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
ACM Transactions on Programming Languages and Systems (TOPLAS)
Solving satisfiability problems with preferences
Constraints
Generalizing the template polyhedral domain
ESOP'11/ETAPS'11 Proceedings of the 20th European conference on Programming languages and systems: part of the joint European conferences on theory and practice of software
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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Extending linear constraints by admitting parameters allows for more abstract problem modeling and reasoning. A lot of focus has been given to conducting research that demonstrates the usefulness of parameterized linear constraints and implementing tools that utilize their modeling strength. However, there is no approach that considers basic theoretical tools related to such constraints that allow for reasoning over them. Hence, in this paper we introduce satisfiability with respect to polyhedral sets and entailment for the class of parameterized linear constraints. In order to study the computational complexities of these problems, we relate them to classes of quantified linear implications. The problem of satisfiability with respect to polyhedral sets is then shown to be co- $\mathbb{NP}$ hard. The entailment problem is also shown to be co- $\mathbb{NP}$ hard in its general form. Nevertheless, we characterize some subclasses for which this problem is in 驴. Furthermore, we examine a weakening and a strengthening extension of the entailment problem. The weak entailment problem is proved to be $\mathbb{NP}$ complete. On the other hand, the strong entailment problem is shown to be co- $\mathbb{NP}$ hard.