Journal Article10.1007/S10472-007-9085-Y
On a decision procedure for quantified linear programs
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TL;DR: It is shown that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties and a taxonomy of quantified linear programs is provided, based on the structure of the quantifier string, to discuss the computational complexities of the constituent classes.
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Abstract: Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes.
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Citations
Mathematical optimization for challenging network planning problems in unbundled liberalized gas markets
Armin Fügenschuh,Björn Geißler,Ralf Gollmer,Christine Hayn,René Henrion,Benjamin Hiller,Jesco Humpola,Thorsten Koch,Thomas Lehmann,Alex Martin,Radoslava Mirkov,Antonio Morsi,Jessica Rövekamp,Lars Schewe,Martin Schmidt,Rüdiger Schultz,Robert Schwarz,Jonas Schweiger,Claudia Stangl,Marc C. Steinbach,Bernhard M. Willert +20 more
TL;DR: The main goal of the research is to develop a numerical solver that is able to solve instances of realistic size of gas network operators, and the main ingredients of the prototypical software implementations are described.
Yasol: An Open Source Solver for Quantified Mixed Integer Programs
Thorsten Ederer,Michael Hartisch,Ulf Lorenz,Thomas Opfer,Jan Wolf +4 more
- 03 Jul 2017
TL;DR: In order to solve the QMIP optimization problem, where the task is to find an especially attractive winning strategy, the problem’s hybrid nature is examined and the open source solver Yasol is presented that combines linear programming techniques with solution techniques from game-tree search.
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Complexity bounds for the controllability of temporal networks with conditions, disjunctions, and uncertainty
TL;DR: In this article, the authors provide tight completeness bounds for strong, weak, and dynamic controllability checking of temporal networks that have conditions, disjunctions, and temporal uncertainty.
10
On quantified linear implications
TL;DR: This paper provides a 2-person game semantics for the QLI problem, which allows us to explore the computational complexities of several of its classes, and proves that the decision problem for QLIs with an arbitrary number of quantifier alternations is PSPACE-hard.
9
On the complexity of quantified linear systems
TL;DR: It is shown that when the authors restrict ourselves to quantified conjunctions of linear inequalities, i.e., quantified linear systems, the complexity classes collapse to polynomial time, which reinforces the importance of sentence formats from the perspective of computational complexity.
8
References
Combinatorial optimization: algorithms and complexity
TL;DR: This clearly written, mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more.
7.6K
A theory of timed automata
Rajeev Alur,David L. Dill +1 more
TL;DR: Alur et al. as discussed by the authors proposed timed automata to model the behavior of real-time systems over time, and showed that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the non-deterministic case and PSPACE-complete in deterministic case.
7.5K
•Book
Theory of Linear and Integer Programming
Alexander Schrijver
- 01 Dec 1986
TL;DR: Introduction and Preliminaries.
Lecture Notes in Artificial Intelligence
P. Brezillon,P. Bouquet +1 more
- 01 Jan 1999
TL;DR: The topics in LNAI include automated reasoning, automated programming, algorithms, knowledge representation, agent-based systems, intelligent systems, expert systems, machine learning, natural-language processing, machine vision, robotics, search systems, knowledge discovery, data mining, and related programming languages.
7.5K
•Book
Integer and Combinatorial Optimization
George L. Nemhauser,Laurence A. Wolsey +1 more
- 01 Jan 1988
TL;DR: This chapter discusses the Scope of Integer and Combinatorial Optimization, as well as applications of Special-Purpose Algorithms and Matching.