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Parameterized Complexity Classification for Interval Constraints

Authors Konrad K. Dabrowski , Peter Jonsson , Sebastian Ordyniak , George Osipov , Marcin Pilipczuk , Roohani Sharma



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Author Details

Konrad K. Dabrowski
  • School of Computing, Newcastle University, UK
Peter Jonsson
  • Department of Computer and Information Science, Linköping University, Sweden
Sebastian Ordyniak
  • School of Computing, University of Leeds, UK
George Osipov
  • Department of Computer and Information Science, Linköping University, Sweden
Marcin Pilipczuk
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
  • IT University Copenhagen, Denmark
Roohani Sharma
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

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Konrad K. Dabrowski, Peter Jonsson, Sebastian Ordyniak, George Osipov, Marcin Pilipczuk, and Roohani Sharma. Parameterized Complexity Classification for Interval Constraints. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 11:1-11:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.11

Abstract

Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most k constraints, where k is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen’s interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set A of 13 basic comparison relations such as "precedes" or "during" for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP(Γ) for all Γ ⊆ A. IA is sometimes extended with unions of the relations in A or first-order definable relations over A, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP(A) in general, we then consider (parameterized) approximation algorithms. We first show that MinCSP(A) cannot be polynomial-time approximated within any constant factor and continue by presenting a factor-2 fpt-approximation algorithm. Once again, this algorithm has its roots in flow augmentation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • (minimum) constraint satisfaction problem
  • Allen’s interval algebra
  • parameterized complexity
  • cut problems

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