Journal of Computer Science and Technology

, Volume 27, Issue 4, pp 668–677 | Cite as

Satisfiability with Index Dependency

  • Hong-Yu LiangEmail author
  • Jing He


We study the Boolean satisfiability problem (SAT) restricted on input formulas for which there are linear arithmetic constraints imposed on the indices of variables occurring in the same clause. This can be seen as a structural counterpart of Schaefer’s dichotomy theorem which studies the SAT problem with additional constraints on the assigned values of variables in the same clause. More precisely, let k-SAT\( \left( {m,\mathcal{A}} \right) \) denote the SAT problem restricted on instances of k-CNF formulas, in every clause of which the indices of the last k − m variables are totally decided by the first m ones through some linear equations chosen from \( \mathcal{A} \). For example, if \( \mathcal{A} \) contains i 3 = i 1 +2i 2 and i 4 = i 2− i 1 +1, then a clause of the input to 4-SAT(2, \( \mathcal{A} \)) has the form yi 1yi 2yi 1 + 2i 2yi 2− i 1 + 1, with y i being x i or \( \overline {xi} \). We obtain the following results: 1) If m ≥ 2, then for any set \( \mathcal{A} \) of linear constraints, the restricted problem k-SAT(m, \( \mathcal{A} \)) is either in P or NP-complete assuming P ≠ NP. Moreover, the corresponding #SAT problem is always #P-complete, and the Max-SAT problem does not allow a polynomial time approximation scheme assuming P  NP. 2) m = 1, that is, in every clause only one index can be chosen freely. In this case, we develop a general framework together with some techniques for designing polynomial-time algorithms for the restricted SAT problems. Using these, we prove that for any \( \mathcal{A} \), #2-SAT(1, \( \mathcal{A} \)) and Max-2-SAT(1, \( \mathcal{A} \)) are both polynomial-time solvable, which is in sharp contrast with the hardness results of general #2-SAT and Max-2-SAT. For fixed k ≥ 3, we obtain a large class of non-trivial constraints \( \mathcal{A} \), under which the problems k-SAT(1, \( \mathcal{A} \)), #k-SAT(1, \( \mathcal{A} \)) and Max-k-SAT(1, \( \mathcal{A} \)) can all be solved in polynomial time or quasi-polynomial time.


Boolean satisfiability problem index-dependency index-width dichotomy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cook S A. The complexity of theorem proving procedures. In Proc. ACM STOC, May 1971, pp.151–158.Google Scholar
  2. [2]
    Aspvall B, Plass M F, Tarjan R E. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett., 1979, 8(3): 121–123.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Valiant L G. The complexity of computing the permanent. Theoret. Comput. Sci., 1979, 8(2): 189–201.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Valiant L G. The complexity of enumeration and reliability problems. SIAM J. Comput., 1979, 8(3): 410–421.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Arora S, Lund C, Motwani R, Sudan M, Szegedy M. Proof verification and the hardness of approximation problems. J. ACM, 1998, 45(3): 501–555.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Håstad J. Some optimal inapproximability results. J. ACM, 2001, 48(4): 798–859.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Henschen L, Wos L. Unit refutations and Horn sets. J. ACM, 1974, 21(4): 590–605.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Yamasaki S, Doshita S. The satisfiability problem for the class consisting of Horn sentences and some non-Horn sentences in propositional logic. Infor. Control, 1983, 59(1–3): 1–12.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Schaefer T J. The complexity of satisfiability problems. In Proc. ACM STOC, May 1978, pp.216–226.Google Scholar
  10. [10]
    Allender E, Bauland M, Immerman N, Schnoor H, Vollmer H. The complexity of satisfiability problems: Refining Schaefer’s theorem. J. Comput. System Sci., 2009, 75(4): 245–254.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Tovey C A. A simplified NP-complete satisfiability problem. Discrete Appl. Math., 1984, 8(1): 85–89.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Kratochvíl J, Savický P, Tuza Z. One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM J. Comput., 1993, 22(1): 203–210.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Gebauer H, Szabó T, Tardos G. The local lemma is tight for SAT. In Proc. the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), Jan. 2011, pp.664–674.Google Scholar
  14. [14]
    Lichtenstein D. Planar formulae and their uses. SIAM J. Comput., 1982, 11(2): 329–343.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Monien B, Sudborough I H. Bandwidth constrained NPcomplete problems. In Proc. ACM STOC, May 1981, pp.207–217.Google Scholar
  16. [16]
    Georgiou K, Papakonstantinou P A. Complexity and algorithms for well-structured k-SAT instances. In Proc. the 11th International Conference on Theory and Applications of Satisfiability Testing (SAT), May 2008, pp.105–118.Google Scholar
  17. [17]
    Rosen K H. Elementary Number Theory and its Applications (5th Edition), Addison Wesley, 2004.Google Scholar
  18. [18]
    Borosh I, Flahive M, Rubin D, Treybig B. A sharp bound for solutions of linear diophantine equations. P. Am. Math. Soc., 1989, 105(4): 844–846.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Borosh I, Flahive M, Treybig B. Small solution of linear Diophantine equations. Discrete Math., 1986, 58(3): 215–220.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Bradley G H. Algorithms for Hermite and Smith normal matrices and linear diophantine equations. Math. Comput., 1971, 25(116): 897–907.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC & Science Press, China 2012

Authors and Affiliations

  1. 1.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

Personalised recommendations