Skip to main content
SpringerLink
Log in

Exact Algorithms for Maximum Two-Satisfiability

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 73 Accesses

Years and Authors of Summarized Original Work

  • 2004; Williams

Problem Definition

In the maximum 2-satisfiability problem (abbreviated as Max 2-Sat), one is given a Boolean formula in conjunctive normal form, such that each clause contains at most two literals. The task is to find an assignment to the variables of the formula such that a maximum number of clauses are satisfied.

Max 2-Sat is a classic optimization problem. Its decision version was proved NP-complete by Garey, Johnson, and Stockmeyer [7], in stark contrast with 2-Sat which is solvable in linear time [2]. To get a feeling for the difficulty of the problem, the NP-completeness reduction is sketched here. One can transform any 3-Sat instance F into a Max 2-Sat instance F′, by replacing each clause of F such as

$$\displaystyle\begin{array}{rcl} c_{i} = (\ell_{1} \vee \ell_{2} \vee \ell_{3}),& & {}\\ \end{array}$$

where â„“1, â„“2, and â„“3 are arbitrary literals, with the collection of 2-CNF clauses

$$\displaystyle\begin{array}{rcl}...

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 1,599.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,999.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Recommended Reading

  1. Alon N, Galil Z, Margalit O (1997) On the exponent of the all-pairs shortest path problem. J Comput Syst Sci 54:255–262

    Article  MathSciNet  MATH  Google Scholar 

  2. Aspvall B, Plass MF, Tarjan RE (1979) A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf Proc Lett 8(3):121–123

    Article  MathSciNet  MATH  Google Scholar 

  3. Bansal N, Raman V (1999) Upper bounds for Max Sat: further improved. In: Proceedings of ISAAC, Chennai. LNCS, vol 1741. Springer, Berlin, pp 247–258

    Google Scholar 

  4. Coppersmith D, Winograd S (1990) Matrix multiplication via arithmetic progressions. JSC 9(3):251–280

    MathSciNet  MATH  Google Scholar 

  5. Dantsin E, Wolpert A (2006) Max SAT for formulas with constant clause density can be solved faster than in O(2n) time. In: Proceedings of the 9th international conference on theory and applications of satisfiability testing, Seattle. LNCS, vol 4121. Springer, Berlin, pp 266–276

    Google Scholar 

  6. Dorn F (2006) Dynamic programming and fast matrix multiplication. In: Proceedings of 14th annual European symposium on algorithms, Zurich. LNCS, vol 4168. Springer, Berlin, pp 280–291

    Google Scholar 

  7. Garey M, Johnson D, Stockmeyer L (1976) Some simplified NP-complete graph problems. Theor Comput Sci 1:237–267

    Article  MathSciNet  MATH  Google Scholar 

  8. Gramm J, Niedermeier R (2000) Faster exact solutions for Max2Sat. In: Proceedings of CIAC. LNCS, vol 1767, Rome. Springer, Berlin, pp 174–186

    Google Scholar 

  9. Hirsch EA (2000) A 2m∕4-time algorithm for Max 2-SAT: corrected version. Electronic colloquium on computational complexity report TR99-036

    Google Scholar 

  10. Itai A, Rodeh M (1978) Finding a minimum circuit in a graph. SIAM J Comput 7(4):413–423

    Article  MathSciNet  MATH  Google Scholar 

  11. Kneis J, Mölle D, Richter S, Rossmanith P (2005) Algorithms based on the treewidth of sparse graphs. In: Proceedings of workshop on graph theoretic concepts in computer science, Metz. LNCS, vol 3787. Springer, Berlin, pp 385–396

    Google Scholar 

  12. Kojevnikov A, Kulikov AS (2006) A new approach to proving upper bounds for Max 2-SAT. In: Proceedings of the seventeenth annual ACM-SIAM symposium on discrete algorithms, Miami, pp 11–17

    MATH  Google Scholar 

  13. Mahajan M, Raman V (1999) Parameterizing above guaranteed values: MAXSAT and MAXCUT. J Algorithms 31(2):335–354

    Article  MathSciNet  MATH  Google Scholar 

  14. Niedermeier R, Rossmanith P (2000) New upper bounds for maximum satisfiability. J Algorithms 26:63–88

    Article  MathSciNet  MATH  Google Scholar 

  15. Scott A, Sorkin G (2003) Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances. In: Proceedings of RANDOM-APPROX 2003, Princeton. LNCS, vol 2764. Springer, Berlin, pp 382–395

    Google Scholar 

  16. Vassilevska Williams V (2012) Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of the 44th annual ACM symposium on theory of computing, New York, pp 887–898

    Google Scholar 

  17. Williams R (2004) On computing k-CNF formula properties. In: Theory and applications of satisfiability testing. LNCS, vol 2919. Springer, Berlin, pp 330–340

    Google Scholar 

  18. Williams R (2005) A new algorithm for optimal 2-constraint satisfaction and its implications. Theor Comput Sci 348(2–3):357–365

    Article  MathSciNet  MATH  Google Scholar 

  19. Williams R (2007) Algorithms and resource requirements for fundamental problems. PhD thesis, Carnegie Mellon University

    Google Scholar 

  20. Woeginger GJ (2003) Exact algorithms for NP-hard problems: a survey. In: Combinatorial optimization – Eureka! You shrink! LNCS, vol 2570. Springer, Berlin, pp 185–207

    Chapter  Google Scholar 

  21. Woeginger GJ (2004) Space and time complexity of exact algorithms: some open problems. In: Proceedings of 1st international workshop on parameterized and exact computation (IWPEC 2004), Bergen. LNCS, vol 3162. Springer, Berlin, pp 281–290

    Google Scholar 

  22. Yuval G (1976) An algorithm for finding all shortest paths using N2. 81 infinite-precision multiplications. Inf Process Lett 4(6):155–156

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Stanford University, Stanford, CA, USA

    Ryan Williams

Authors
  1. Ryan Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ryan Williams .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media New York

About this entry

Cite this entry

Williams, R. (2016). Exact Algorithms for Maximum Two-Satisfiability. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_227

Download citation

Publish with us

Policies and ethics

Search

Navigation