Acta Informatica

, Volume 53, Issue 2, pp 149–170 | Cite as

Optimization in temporal qualitative constraint networks

  • Jean-François Condotta
  • Souhila Kaci
  • Yakoub Salhi
Original Article


Various formalisms for representing and reasoning about temporal information with qualitative constraints have been studied in the past three decades. The most known are definitely the Point Algebra \((\mathsf {PA})\) and the Interval Algebra \(({\mathsf {IA}})\) proposed by Allen. In this paper, for both calculi, we study a problem that we call the minimal consistency problem \((\mathsf {MinCons})\). Given a temporal qualitative constraint network \((\mathsf {TQCN})\) and a positive integer \(k\), this problem consists in deciding whether or not this \(\mathsf {TQCN}\) admits a solution using at most \(k\) distinct points on the line.We show that \(\mathsf {MinCons}\) for \(\mathsf {PA}\) can be encoded into the finitary versions of Gödel logic. Furthermore, we prove that the \(\mathsf {MinCons}\) problem is \(\mathsf {NP}\)-complete for both \(\mathsf {PA}\) and \({\mathsf {IA}}\), in the general case. However, we show that for \(\mathsf {TQCN}\)s defined on the convex relations, \(\mathsf {MinCons}\) is polynomial. For such \(\mathsf {TQCN}\)s, we give a polynomial method that allows one to obtain compact scenarios.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Jean-François Condotta
    • 1
  • Souhila Kaci
    • 2
  • Yakoub Salhi
    • 1
  1. 1.CRIL CNRS, UMR 8188Université Lille-Nord de FranceLensFrance
  2. 2.LIRMM CNRS, UMR 5506Université MontpellierMontpellierFrance

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