Integer programming with 2-variable equations and 1-variable inequalities

doi:10.1016/j.ipl.2009.01.025

Information Processing Letters

Volume 109, Issue 11, 16 May 2009, Pages 572-575

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Abstract

We present an efficient algorithm to find an optimal integer solution of a given system of 2-variable equalities and 1-variable inequalities with respect to a given linear objective function. Our algorithm has worst-case running time in $O (N^{2})$ where N is the number of bits in the input.

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There are more references available in the full text version of this article.

Cited by (4)

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Temporal reasoning problems arise in many areas of AI, including planning, natural language understanding, and reasoning about physical systems. The computational complexity of continuous-time temporal constraint reasoning is fairly well understood. There are, however, many different cases where discrete time must be considered; various scheduling problems and reasoning about sampled physical systems are two examples. Here, the complexity of temporal reasoning is not as well-studied nor as well-understood. In order to get a better understanding, we consider the powerful Horn disjunctive linear relations (Horn DLR) formalism adapted for discrete time and study its computational complexity. We show that the full formalism is NP-hard and identify several maximal tractable subclasses. We also ‘lift’ the maximality results to obtain hardness results for other families of constraints. Finally, we discuss how the results and techniques presented in this paper can be used for studying even more expressive classes of temporal constraints.
On polynomial kernels for sparse integer linear programs
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Discrete-time temporal reasoning with horn DLRs
2011, IJCAI International Joint Conference on Artificial Intelligence
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2009, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

¹: Partially supported by the Swedish Research Council (VR) under grant 2008-4675 and the Swedish–French foundation.

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Integer programming with 2-variable equations and 1-variable inequalities

Abstract

Linear Algebra Appl.

Algorithms for the solution of systems of linear Diophantine equations

SIAM J. Comput.

A linear algorithm for integer programming in the plane

Math. Programming

Polynomial algorithms for computing the smith and Hermite normal forms of an integer matrix

SIAM J. Comput.

Integer programming with a fixed number of variables

Math. Oper. Res.