Theory of Computing Systems

, Volume 59, Issue 1, pp 76–98 | Cite as

Complexity of Approximating CSP with Balance / Hard Constraints

  • Venkatesan Guruswami
  • Euiwoong LeeEmail author


We study two natural extensions of Constraint Satisfaction Problems (CSPs). Balance-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of Hard-Max-CSP consists of soft constraints and hard constraints, and the goal is to maximize the weight of satisfied soft constraints while satisfying all the hard constraints. These two extensions contain many fundamental problems not captured by CSPs, and challenge traditional theories about CSPs in a more general framework. Max-2-SAT and Max-Horn-SAT are the only two nontrivial classes of Boolean CSPs that admit a robust satisfibiality algorithm, i.e., an algorithm that finds an assignment satisfying at least (1 − g(ε)) fraction of constraints given a (1 − ε)-satisfiable instance, where g(ε) → 0 as ε → 0, and g(0) = 0. We prove the inapproximability of these problems with balance or hard constraints, showing that each variant changes the nature of the problems significantly (in different ways). For instance, deciding whether an instance of 2-SAT admits a balanced assignment is NP-hard, and for Max-2-SAT with hard constraints, it is hard to find a constant-factor approximation even on (1 − ε)-satisfiable instances (in particular, the version with hard constraints does not admit a robust satisfiability algorithm). We also study hardness results for a certain CSP over a larger domain capturing ordering constraints: we show that hard constraints rule out constant-factor approximation algorithms. All our hardness results are almost optimal — they completely rule out algorithms with certain properties, or can be matched by simple extensions to existing algorithms.


Constraint satisfaction problems Complexity dichotomy Approximation algorithms Hardness of approximation Unique games hardness 



Venkatesan Guruswami is supported in part by a Packard Fellowship and NSF grant CCF-1115525 and Euiwoong Lee is supported by a Samsung Fellowship, MSR-CMU Center for Computational Thinking, and NSF CCF-1115525. We thank Andrei Krokhin for raising the question of whether Hard-2-SAT admits a robust satisfiability algorithm, which was the impetus for this work, and also for several useful comments on the write-up. We also thank Per Austrin for useful discussions.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA

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