Skip to main content
SpringerLink
Log in

Inclusion/Exclusion Meets Measure and Conquer

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times.

In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number.

This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Algorithm 2
Algorithm 3
Algorithm 4
Overview 1
Fig. 1

Similar content being viewed by others

Notes

  1. An algorithm similar to Algorithm 4 has been published in an earlier draft of a part of this paper [69]. We note that the analysis published in [69] contains a mistake. We corrected this mistake and as a result, the preference order that can be found in the algorithm in [69] is no longer required. For the rest, both algorithms are very similar though they appear different. This is because Algorithm 4 is presented using red-blue dominating sets, while the Algorithm in [69] uses an equivalent presentation involving set covers. Furthermore, two reduction rules are omitted from Algorithm 4: one rule is superfluous (the base case in [69] is covered by the procedure CountRBDS2-DP(G,m)), and the other rule (connected components) is never used in the analysis and could thus be omitted.

  2. Gaspers et al. [41] did not use Proposition 5 (see [68]) but slower algorithms on tree decompositions instead.

References

  1. Amini, O., Fomin, F.V., Saurabh, S.: Implicit branching and parameterized partial cover problems. J. Comput. Syst. Sci. 77(6), 1159–1171 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Amini, O., Fomin, F.V., Saurabh, S.: Counting subgraphs via homomorphisms. SIAM J. Discrete Math. 26(2), 695–717 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andersson, G., Engebretsen, L.: Better approximation algorithms for SET SPLITTING and NOT-ALL-EQUAL SAT. Inf. Process. Lett. 65(6), 305–311 (1998)

    Article  MathSciNet  Google Scholar 

  4. Bax, E.T.: Inclusion and exclusion algorithm for the Hamiltonian path problem. Inf. Process. Lett. 47(4), 203–207 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Binkele-Raible, D., Fernau, H.: Enumerate & measure: improving parameter budget management. In: Raman, V., Saurabh, S. (eds.) 5th International Symposium on Parameterized and Exact Computation, IPEC 2010. Lecture Notes in Computer Science, vol. 6478, pp. 38–49. Springer, Berlin (2010)

    Google Scholar 

  6. Björklund, A.: Determinant sums for undirected Hamiltonicity. In: 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, pp. 173–182. IEEE Computer Society, New York (2010)

    Chapter  Google Scholar 

  7. Björklund, A.: Exact covers via determinants. In: Marion, J.-Y., Schwentick, T. (eds.) 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010. Leibniz International Proceedings in Informatics, vol. 3, pp. 95–106. Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik (2010)

    Google Scholar 

  8. Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Johnson, D.S., Feige, U. (eds.) 39th Annual ACM Symposium on Theory of Computing, STOC 2007, pp. 67–74. ACM Press, New York (2007)

    Google Scholar 

  10. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Counting paths and packings in halves. In: Fiat, A., Sanders, P. (eds.) 17th Annual European Symposium on Algorithms, ESA 2009. Lecture Notes in Computer Science, vol. 5757, pp. 578–586. Springer, Berlin (2009)

    Chapter  Google Scholar 

  11. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed Moebius inversion and graphs of bounded degree. Theory Comput. Syst. 47(3), 637–654 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Covering and packing in linear space. Inf. Process. Lett. 111(21–22), 1033–1036 (2011)

    Article  MATH  Google Scholar 

  13. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)

    Article  MathSciNet  Google Scholar 

  16. Bourgeois, N., Croce, F.D., Escoffier, B., Paschos, V.T.: Algorithms for dominating clique problems. Theor. Comput. Sci. 459, 77–88 (2012)

    Article  MATH  Google Scholar 

  17. Bourgeois, N., Escoffier, B., Paschos, V.T.: Fast algorithms for min independent dominating set. In: Patt-Shamir, B., Ekim, T. (eds.) 17th International Colloquium Structural Information and Communication Complexity, SIROCCO 2010. Lecture Notes in Computer Science, vol. 6058, pp. 247–261. Springer, Berlin (2010)

    Google Scholar 

  18. Bourgeois, N., Escoffier, B., Paschos, V.T., van Rooij, J.M.M.: Maximum independent set in graphs of average degree at most three in O(1.08537n). In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) 7th Annual Conference on Theory and Applications of Models of Computation, TAMC 2010. Lecture Notes in Computer Science, vol. 6108, pp. 373–384. Springer, Berlin (2010)

    Google Scholar 

  19. Bourgeois, N., Escoffier, B., Paschos, V.T., van Rooij, J.M.M.: Fast algorithms for max independent set. Algorithmica 62(1–2), 382–415 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chen, J., Lu, S.: Improved parameterized set splitting algorithms: a probabilistic approach. Algorithmica 54(4), 472–489 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cygan, M., Dell, H., Lokshtanov, D., Marx, D., Nederlof, J., Okamoto, Y., Paturi, R., Saurabh, S., Wahlström, M.: On problems as hard as CNF-SAT. In: IEEE Conference on Computational Complexity, pp. 74–84. IEEE, New York (2012)

    Google Scholar 

  22. Dehne, F.K.H.A., Fellows, M.R., Fernau, H., Prieto, E., Rosamond, F.A.: Nonblocker: parameterized algorithmics for minimum dominating set. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Stuller, J. (eds.) 32nd Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2006. Lecture Notes in Computer Science, vol. 3831, pp. 237–245. Springer, Berlin (2006)

    Chapter  Google Scholar 

  23. Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A.: An FPT algorithm for set splitting. In: Bodlaender, H.L. (ed.) 29th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2003. Lecture Notes in Computer Science, vol. 2880, pp. 180–191. Springer, Berlin (2003)

    Google Scholar 

  24. Dehne, F.K.H.A., Fellows, M.R., Rosamond, F.A., Shaw, P.: Greedy localization, iterative compression, modeled crown reductions: new FPT techniques, an improved algorithm for set splitting, and a novel 2k kernelization for vertex cover. In: Downey, R.G., Fellows, M.R., Dehne, F.K.H.A. (eds.) 1st International Workshop on Parameterized and Exact Computation, IWPEC 2004. Lecture Notes in Computer Science, vol. 3162, pp. 271–280. Springer, Berlin (2004)

    Google Scholar 

  25. Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congr. Numer. 87, 161–178 (1992)

    MathSciNet  Google Scholar 

  26. Eppstein, D.: Quasiconvex analysis of multivariate recurrence equations for backtracking algorithms. ACM Trans. Algorithms 2(4), 492–509 (2006)

    Article  MathSciNet  Google Scholar 

  27. Erdös, P.: On a combinatorial problem, I. Nord. Mat. Tidskrift 11, 5–10 (1963)

    MATH  Google Scholar 

  28. Erdös, P.: On a combinatorial problem, II. Acta Math. Hung. 15(3), 445–447 (1964)

    Article  MATH  Google Scholar 

  29. Fernau, H., Kneis, J., Kratsch, D., Langer, A., Liedloff, M., Raible, D., Rossmanith, P.: An exact algorithm for the maximum leaf spanning tree problem. Theor. Comput. Sci. 412(45), 6290–6302 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fomin, F.V., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. Algorithmica 52(2), 153–166 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56(5) (2009)

  34. Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1) (2008)

  35. Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1) (2008)

  36. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  37. Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovic, J., Nagl, M., Westfechtel, B. (eds.) 30th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2004. Lecture Notes in Computer Science, vol. 3353, pp. 24–256. Springer, Berlin (2004)

    Google Scholar 

  38. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Subexponential algorithms for partial cover problems. Inf. Process. Lett. 111(16), 814–818 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  40. Gaspers, S., Kratsch, D., Liedloff, M.: On independent sets and bicliques in graphs. Algorithmica 62(3–4), 637–658 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  41. Gaspers, S., Kratsch, D., Liedloff, M., Todinca, I.: Exponential time algorithms for the minimum dominating set problem on some graph classes. ACM Trans. Algorithms 6(1) (2009)

  42. Gaspers, S., Liedloff, M.: A branch-and-reduce algorithm for finding a minimum independent dominating set. Discrete Math. Theor. Comput. Sci. 14(1), 29–42 (2012)

    MATH  MathSciNet  Google Scholar 

  43. Gaspers, S., Sorkin, G.B.: A universally fastest algorithm for max 2-Sat, max 2-CSP, and everything in between. J. Comput. Syst. Sci. 78(1), 305–335 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  44. Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory, Ser. B 16(1), 47–56 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  45. Grandoni, F.: Exact algorithms for hard graph problems. PhD thesis, Department of Computer Science, Systems and Production, Universitá degli Studi di Roma “Tor Vergata”, Rome, Italy (2004)

  46. Grandoni, F.: A note on the complexity of minimum dominating set. J. Discrete Algorithms 4(2), 209–214 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  47. Iwata, Y.: A faster algorithm for dominating set analyzed by the potential method. In: Marx, D., Rossmanith, P. (eds.) IPEC. Lecture Notes in Computer Science, vol. 7112, pp. 41–54. Springer, Berlin (2011)

    Google Scholar 

  48. Karp, R.M.: Dynamic programming meets the principle of inclusion-exclusion. Oper. Res. Lett. 1(2), 49–51 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  49. Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: A bound on the pathwidth of sparse graphs with applications to exact algorithms. SIAM J. Discrete Math. 23(1), 407–427 (2009)

    Article  MATH  Google Scholar 

  50. Kneis, J., Mölle, D., Rossmanith, P.: Partial vs. complete domination: t-dominating set. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plasil, F. (eds.) 33rd Conference on Current Trends in Theory and Practice of Computer Science, SOFSEM 2007. Lecture Notes in Computer Science, vol. 4362, pp. 367–376. Springer, Berlin (2007)

    Chapter  Google Scholar 

  51. Kohn, S., Gottlieb, A., Kohn, M.: A generating function approach to the traveling salesman problem. In: Proceedings of the 1977 Annual Conference of the ACM, pp. 294–300. ACM Press, New York (1977)

    Chapter  Google Scholar 

  52. Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) 36th International Colloquium on Automata, Languages and Programming (1), ICALP 2009. Lecture Notes in Computer Science, vol. 5555, pp. 653–664. Springer, Berlin (2009)

    Google Scholar 

  53. Liedloff, M.: Algorithmes exacts et exponentiels pour les problèmes NP-difficiles: domination, variantes et généralisation. PhD thesis, Laboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine, Metz, France (2007)

  54. Lokshtanov, D., Saurabh, S.: Even faster algorithm for set splitting! In: Chen, J., Fomin, F.V. (eds.) 4th International Workshop on Parameterized and Exact Computation, IWPEC 2009. Lecture Notes in Computer Science, vol. 5917, pp. 288–299. Springer, Berlin (2009)

    Google Scholar 

  55. Lokshtanov, D., Sloper, C.: Fixed parameter set splitting, linear kernel and improved running time. In: Broersma, H., Johnson, M., Szeider, S. (eds.) 1st Algorithms and Complexity in Durham Workshop, ACiD 2005. Texts in Algorithmics, vol. 4, pp. 105–113. King’s College, London (2005)

    Google Scholar 

  56. Lovász, L.: Coverings and colorings of hypergraphs. Congr. Numer. 8, 3–12 (1973)

    Google Scholar 

  57. Nederlof, J.: Space and time efficient structural improvements of dynamic programming algorithms. PhD thesis, Department of Informatics, University of Bergen, Bergen, Norway (2011)

  58. Nederlof, J.: Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 1–17 (2012)

  59. Paulusma, D., van Rooij, J.M.M.: On partitioning a graph into two connected subgraphs. Theor. Comput. Sci. 412(48), 6761–6769 (2011)

    Article  MATH  Google Scholar 

  60. Radhakrishnan, J., Srinivasan, A.: Improved bounds and algorithms for hypergraph 2-coloring. Random Struct. Algorithms 16(1), 4–32 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  61. Riege, T., Rothe, J.: An exact 2.9416n algorithm for the three domatic number problem. In: Jedrzejowicz, J., Szepietowski, A. (eds.) 30th International Symposium on Mathematical Foundations of Computer Science, MFCS 2005. Lecture Notes in Computer Science, vol. 3618, pp. 733–744. Springer, Berlin (2005)

    Chapter  Google Scholar 

  62. Riege, T., Rothe, J., Spakowski, H., Yamamoto, M.: An improved exact algorithm for the domatic number problem. Inf. Process. Lett. 101(3), 101–106 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  63. Schiermeyer, I.: Efficiency in exponential time for domination-type problems. Discrete Appl. Math. 156(17), 3291–3297 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  64. Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–579 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  65. van Rooij, J.M.M.: Exact exponential-time algorithms for domination problems in graphs. PhD thesis, Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2011)

  66. van Rooij, J.M.M., Bodlaender, H.L.: Exact algorithms for dominating set. Discrete Appl. Math. 159(17), 2147–2164 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  67. van Rooij, J.M.M., Bodlaender, H.L.: Exact algorithms for edge domination. Algorithmica 64(4), 535–563 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  68. van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic programming on tree decompositions using generalised fast subset convolution. In: Fiat, A., Sanders, P. (eds.) 17th Annual European Symposium on Algorithms, ESA 2009. Lecture Notes in Computer Science, vol. 5757, pp. 566–577. Springer, Berlin (2009)

    Chapter  Google Scholar 

  69. van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/exclusion meets measure and conquer. In: Fiat, A., Sanders, P. (eds.) 17th Annual European Symposium on Algorithms, ESA 2009. Lecture Notes in Computer Science, vol. 5757, pp. 554–565. Springer, Berlin (2009)

    Chapter  Google Scholar 

  70. Zhang, J., Ye, Y., Han, Q.: Improved approximations for max set splitting and max NAE SAT. Discrete Appl. Math. 142(1–3), 133–149 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  71. Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: 31th Annual ACM Symposium on Theory of Computing, STOC 1999, pp. 679–687. ACM Press, New York (1999)

    Google Scholar 

Download references

Acknowledgements

We thank Hans L. Bodlaender for his guidance and enthusiasm during this research.

Author information

Authors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Jesper Nederlof

  2. Consultants in Quantitative Methods, Vonderweg 16, 5616 RM, Eindhoven, The Netherlands

    Johan M. M. van Rooij

  3. Lehrstuhl für Informatik I, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Thomas C. van Dijk

Authors
  1. Jesper Nederlof
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johan M. M. van Rooij
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas C. van Dijk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesper Nederlof.

Additional information

Preliminary parts of this paper have appeared in the conference proceedings of three different conferences. Preliminary parts appeared under the title ‘Inclusion/Exclusion Meets Measure and Conquer: Exact algorithms for counting dominating sets’ at the 17th Annual European Symposium on Algorithms (ESA 2009), Lecture Notes in Computer Science 5757, pp. 554–565, under the title ‘Polynomial Space Algorithms for Counting Dominating Sets and the Domatic Number’ at the 7th International Conference on Algorithms and Complexity (CIAC 2010), Lecture Notes in Computer Science 6078, pp. 73–84, and under the title ‘Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting’ at the 5th International Symposium on Parameterized and Exact Computation (IPEC 2011), Lecture Notes in Computer Science 6478, pp. 204–215. The last paper received the best student paper award of IPEC 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nederlof, J., van Rooij, J.M.M. & van Dijk, T.C. Inclusion/Exclusion Meets Measure and Conquer. Algorithmica 69, 685–740 (2014). https://doi.org/10.1007/s00453-013-9759-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-013-9759-2

Keywords

Search

Navigation