Is Constraint Satisfaction Over Two Variables Always Easy?

  • Lars Engebretsen
  • Venkatesan Guruswami
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)


By the breakthrough work of Håstad, several constraint satisfaction problems are now known to have the following approximation resistance property: satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm, we know that every Boolean 2-CSP has a non-trivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 - x1/d 2).

In an attempt to make progress towards this question, in this paper we prove, firstly, that a slight restriction of this problem, namely a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, secondly, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a non-trivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold.


Approximation Algorithm Constraint Satisfaction Constraint Satisfaction Problem Performance Ratio Hardness Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Lars Engebretsen
    • 1
  • Venkatesan Guruswami
    • 2
  1. 1.Department of Numerical Analysis and Computer ScienceRoyal Institute of TechnologyStockholmSweden
  2. 2.Miller Institute for Basic Research in ScienceUniversity of California at BerkeleyBerkeleyUSA

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