New ${\bf \frac{3}{4}}$-Approximation Algorithms for the Maximum Satisfiability Problem
SIAM Journal on Discrete Mathematics
Finding almost-satisfying assignments
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
The Approximability of Constraint Satisfaction Problems
SIAM Journal on Computing
Improved Approximation Algorithms for MAX k-CUT and MAX BISECTION
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
The RPR2 Rounding Technique for Semidefinite Programs
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
SIAM Journal on Computing
Near-optimal algorithms for maximum constraint satisfaction problems
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Nonuniform Boolean constraint satisfaction problems with cardinality constraint
ACM Transactions on Computational Logic (TOCL)
Constraint satisfaction problems and global cardinality constraints
Communications of the ACM
Inapproximability of hypergraph vertex cover and applications to scheduling problems
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Approximating CSPs with global cardinality constraints using SDP hierarchies
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Improved approximation algorithms for MAX NAE-SAT and MAX SAT
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
Robust Satisfiability for CSPs: Hardness and Algorithmic Results
ACM Transactions on Computation Theory (TOCT)
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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).