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An SMT Approach to Fractional Hypertree Width

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Principles and Practice of Constraint Programming (CP 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11008))

Abstract

Bounded fractional hypertree width () is the most general known structural property that guarantees polynomial-time solvability of the constraint satisfaction problem. Bounded generalizes other structural properties like bounded induced width and bounded hypertree width.

We propose, implement and test the first practical algorithm for computing the and its associated structural decomposition. We provide an extensive empirical evaluation of our method on a large class of benchmark instances which also provides a comparison with known exact decomposition methods for hypertree width. Our approach is based on an efficient encoding of the decomposition problem to SMT (SAT modulo Theory) with Linear Arithmetic as implemented in the SMT solver . The encoding is further strengthened by preprocessing and symmetry breaking methods. Our experiments show (i) that  can indeed be computed exactly for a wide range of benchmark instances, and (ii) that state-of-the art SMT techniques can be successfully applied for structural decomposition.

The work has been supported by the Austrian Science Fund (FWF), Grants Y698 and P26696, and the German Science Fund (DFG), Grant HO 1294/11-1. Fichte and Hecher are also affiliated with the University of Potsdam, Germany.

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Notes

  1. 1.

    See: github.com/daajoe/frasmt.

  2. 2.

    See: Benchmark repository [16] and results/raw data [17].

  3. 3.

    See: https://conda.io/docs/user-guide/install/download.html.

  4. 4.

    github.com/daajoe/detkdecomp.

  5. 5.

    https://www.dbai.tuwien.ac.at/proj/hypertree/benchmarks.zip.

  6. 6.

    See: spectreattack.com.

References

  1. Alviano, M., Dodaro, C.: Anytime answer set optimization via unsatisfiable core shrinking. Theory Pract. Log. Program. 16(5–6), 533–551 (2016)

    Article  MathSciNet  Google Scholar 

  2. Arocena, P.C., Glavic, B., Ciucanu, R., Miller, R.J.: The iBench integration metadata generator. In: Li, C., Markl, V. (eds.) Proceedings of Very Large Data Bases (VLDB) Endowment, vol. 9:3, pp. 108–119. VLDB Endowment, November 2015. https://github.com/RJMillerLab/ibench

    Article  Google Scholar 

  3. Audemard, G., Boussemart, F., Lecoutre, C., Piette, C.: XCSP3: An XML-Based Format Designed to Represent Combinatorial Constrained Problems. http://xcsp.org (2016)

  4. Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. In: Iliopoulos, C.S., Pissis, S.P., Puglisi, S.J., Raman, R. (eds.) 16th International Symposium on Experimental Algorithms, SEA 2017, 21–23 June 2017, London, UK, LIPIcs, vol. 75, pp. 28:1–28:21. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)

    Google Scholar 

  5. Benedikt, M., Konstantinidis, G., Mecca, G., Motik, B., Papotti, P., Santoro, D., Tsamoura, E.: Benchmarking the chase. In: Geerts, F. (ed.) Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS 2017), pp. 37–52. Association for Computing Machinery, New York, Chicago (2017). https://github.com/dbunibas/chasebench

  6. Berg, J., Lodha, N., Järvisalo, M., Szeider., S.: MaxSAT benchmarks based on determining generalized hypertree-width. Technical report, MaxSAT Evaluation 2017 (2017)

    Google Scholar 

  7. Berg, J., Järvisalo, M.: SAT-based approaches to treewidth computation: an evaluation. In: 26th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2014, Limassol, Cyprus, 10–12 November 2014, pp. 328–335. IEEE Computer Society (2014)

    Google Scholar 

  8. Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209(1–2), 1–45 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6(2), 181–188 (1993)

    Article  MathSciNet  Google Scholar 

  10. Carbonnel, C., Cooper, M.C.: Tractability in constraint satisfaction problems: a survey. Constraints 21(2), 115–144 (2016)

    Article  MathSciNet  Google Scholar 

  11. Cohen, D., Jeavons, P., Gyssens, M.: A unified theory of structural tractability for constraint satisfaction problems. J. Comput. Syst. Sci. 74(5), 721–743 (2008)

    Article  MathSciNet  Google Scholar 

  12. Committee, M.S.: IEEE standard for floating-point arithmetic. IEEE Std 754-2008, pp. 1–70, August 2008

    Google Scholar 

  13. Dechter, R.: Tractable structures for constraint satisfaction problems. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, vol. I, chap. 7, pp. 209–244. Elsevier (2006)

    Google Scholar 

  14. Dell, H., Komusiewicz, C., Talmon, N., Weller, M.: The PACE 2017 parameterized algorithms and computational experiments challenge: the second iteration. In: Lokshtanov, D., Nishimura, N. (eds.) Proceedings of the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), pp. 30:1–30:13. LIPIcs (2017)

    Google Scholar 

  15. Durand, A., Mengel, S.: Structural tractability of counting of solutions to conjunctive queries. Theoret. Comput. Sci. 57(4), 1202–1249 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Fichte, J.K., Hecher, M., Lodha, N., Szeider, S.: A Benchmark Collection of Hypergraphs, June 2018. https://doi.org/10.5281/zenodo.1289383

  17. Fichte, J.K., Hecher, M., Lodha, N., Szeider, S.: Analyzed Benchmarks and Raw Data on Experiments for FraSMT, June 2018. https://doi.org/10.5281/zenodo.1289429

  18. Fichte, J.K., Lodha, N., Szeider, S.: SAT-based local improvement for finding tree decompositions of small width. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 401–411. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_25

    Chapter  MATH  Google Scholar 

  19. Fischl, W., Gottlob, G., Longo, D.M., Pichler, R.: HyperBench: A Benchmark of Hypergraphs (2017). http://hyperbench.dbai.tuwien.ac.at

  20. Fischl, W., Gottlob, G., Pichler, R.: Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI symposium on principles of database systems. In: den Bussche, J.V., Arenas, M. (eds.) Conference SIGMOD/PODS 2018 International Conference on Management of Data, Houston, TX, USA, 10–15 June 2018, pp. 17–32. ACM (2018)

    Google Scholar 

  21. Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 29(1), 24–32 (1982)

    Article  Google Scholar 

  22. Gebser, M., Kaminski, R., Kaufmann, B., Schaub, T.: Multi-shot ASP Solving with Clingo. CoRR abs/1705.09811 (2017). http://arxiv.org/abs/1705.09811

  23. Geerts, F., Mecca, G., Papotti, P., Santoro, D.: Mapping and cleaning. In: Cruz, I., Ferrari, E., Tao, Y. (eds.) Proceedings of the IEEE 30th International Conference on Data Engineering (ICDE 2014), pp. 232–243, March 2014

    Google Scholar 

  24. Gottlob, G., Leone, N., Scarcello, F.: Hypertree decompositions and tractable queries. J. Comput. Syst. Sci. 64(3), 579–627 (2002)

    Article  MathSciNet  Google Scholar 

  25. Gottlob, G., Leone, N., Scarcello, F.: On tractable queries and constraints. In: Bench-Capon, T.J.M., Soda, G., Tjoa, A.M. (eds.) DEXA 1999. LNCS, vol. 1677, pp. 1–15. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48309-8_1

    Chapter  Google Scholar 

  26. Gottlob, G., Samer, M.: A backtracking-based algorithm for hypertree decomposition. J. Exp. Alg. 13, 1:1.1–1:1.19 (2009)

    Article  MathSciNet  Google Scholar 

  27. Grohe, M., Marx, D.: Constraint solving via fractional edge covers. In: Proceedings of the of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 289–298. ACM Press (2006)

    Google Scholar 

  28. Grohe, M., Marx, D.: Constraint solving via fractional edge covers. ACM Trans. Alg. 11(1) (2014). Article 4, 20

    Google Scholar 

  29. Guo, Y., Pan, Z., Heflin, J.: LUBM Benchmark OWL Knowl. Base Syst. Web semantics: science, services and agents on the world wide web 3(2), 158–182 (2005)

    Article  Google Scholar 

  30. Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using networkx. In: Gäel Varoquaux, T.V., Millman, J. (eds.) Proceedings of the 7th Python in Science Conference (SciPy 2008), Pasadena, CA, USA, pp. 11–15, August 2008

    Google Scholar 

  31. Kaminski, R., Schneider, M., Rabener, T., et al.: Benchmark-Tool (2017). https://github.com/potassco/benchmark-tool

  32. Khamis, M.A., Ngo, H.Q., Rudra, A.: FAQ: questions asked frequently. In: Milo, T., Tan, W. (eds.) Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016, San Francisco, CA, USA, June 26–July 01 2016. pp. 13–28. Association for Computer Machinery, New York (2016)

    Google Scholar 

  33. Khamis, M.A., Ngo, H.Q., Rudra, A.: FAQ: questions asked frequently. CoRR abs/1504.04044 (2017). http://arxiv.org/abs/1504.04044v6. Full version of [32]

  34. Leis, V., Gubichev, A., Mirchev, A., Boncz, P., Kemper, A., Neumann, T.: How good are query optimizers, really? Proc. Very Large Data Bases (VLDB) Endow. 9(3), 204–215 (2015)

    Article  Google Scholar 

  35. Lodha, N., Ordyniak, S., Szeider, S.: A SAT approach to branchwidth. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 179–195. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_12

    Chapter  MATH  Google Scholar 

  36. Lodha, N., Ordyniak, S., Szeider, S.: SAT-encodings for special treewidth and pathwidth. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 429–445. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_27

    Chapter  MATH  Google Scholar 

  37. Marx, D.: Approximating fractional hypertree width. TALG 6(2) (2010). Article 17, 29

    Article  MathSciNet  Google Scholar 

  38. de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

    Chapter  Google Scholar 

  39. Rose, D.J.: On simple characterizations of \(k\)-trees. Discrete Math. 7, 317–322 (1974)

    Article  MathSciNet  Google Scholar 

  40. van Rossum, G.: Python tutorial. CS-R9526, Centrum voor Wiskunde en Informatica (CWI), Amsterdam, May 1995

    Google Scholar 

  41. Roussel, O.: Controlling a solver execution with the runsolver tool. J. Satisfiability Boolean Model. Comput. 7, 139–144 (2011)

    MathSciNet  MATH  Google Scholar 

  42. Samer, M., Veith, H.: Encoding treewidth into SAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 45–50. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_6

    Chapter  MATH  Google Scholar 

  43. Transaction Processing Performance Council (TPC): TPC-H decision support benchmark. Technical report, TPC (2014). http://www.tpc.org/tpch/default.asp

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Authors and Affiliations

  1. International Center of Computational Logic, TU Dresden, Dresden, Germany

    Johannes K. Fichte

  2. Database and Artificial Intelligence Group, TU Wien, Vienna, Austria

    Markus Hecher

  3. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Neha Lodha & Stefan Szeider

Authors
  1. Johannes K. Fichte
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  2. Markus Hecher
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  3. Neha Lodha
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  4. Stefan Szeider
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Corresponding authors

Correspondence to Johannes K. Fichte , Markus Hecher , Neha Lodha or Stefan Szeider .

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  1. Carnegie Mellon University, Pittsburgh, PA, USA

    John Hooker

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Fichte, J.K., Hecher, M., Lodha, N., Szeider, S. (2018). An SMT Approach to Fractional Hypertree Width. In: Hooker, J. (eds) Principles and Practice of Constraint Programming. CP 2018. Lecture Notes in Computer Science(), vol 11008. Springer, Cham. https://doi.org/10.1007/978-3-319-98334-9_8

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