A Join-Based Hybrid Parameter for Constraint Satisfaction
We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP.
Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances.
- 2.Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pp. 593–603 (2002)Google Scholar
- 8.Cooper, M.C., Zivny, S.: Hybrid tractable classes of constraint problems. In: The Constraint Satisfaction Problem, volume 7 of Dagstuhl Follow-Ups, pp. 113–135. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
- 10.Deville, Y., Van Hentenryck, P.: An efficient arc consistency algorithm for a class of CSP problems. In: Proceedings of the 12th International Joint Conference on Artificial Intelligence, Sydney, Australia, pp. 325–330, 24–30 August 1991Google Scholar
- 13.Fischl, W., Gottlob, G., Pichler, R.: General and fractional hypertree decompositions: hard and easy cases. In: Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, Houston, TX, USA, pp. 17–32. ACM, 10–15 June 2018Google Scholar
- 14.Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science, vol. XIV. An EATCS Series. Springer, Berlin (2006). https://doi.org/10.1007/3-540-29953-X
- 19.Grohe, M.: Logic, graphs, and algorithms. In: Logic and Automata: History and Perspectives. Texts in Logic and Games, vol. 2, pp. 357–422. Amsterdam University Press (2007)Google Scholar
- 20.Grohe, M., Marx, D.: Constraint solving via fractional edge covers. ACM Trans. Algorithms 11(1), 4:1–4:20 (2014). http://doi.acm.org/10.1145/2636918
- 22.Ioannidis, Y.E., Kang, Y.C.: Left-deep vs. bushy trees: an analysis of strategy spaces and its implications for query optimization. In: Proceedings of the 1991 ACM SIGMOD International Conference on Management of Data, pp. 168–177. ACM Press (1991)Google Scholar
- 23.Khamis, M.A., Ngo, H.Q., Rudra, A.: FAQ: questions asked frequently. In: Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016, San Francisco, CA, USA, pp. 3–28. ACM, 26 June–01 July 2016Google Scholar
- 24.Khamis, M.A., Ngo, H.Q., Suciu, D.: What do Shannon-type inequalities, submodular width, and disjunctive datalog have to do with one another? In: Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2017, Chicago, IL, USA, pp. 429–444. ACM, 14–19 May 2017Google Scholar
- 33.Yannakakis, M.: Algorithms for acyclic database schemes. In: Proceedings of 7th International Conference Very Large Data Bases, Cannes, France, pp. 81–94. IEEE Computer Society, 9–11 September 1981Google Scholar