One more occurrence of variables makes satisfiability jump from trivial to NP-complete
SIAM Journal on Computing
A dichotomy theorem for maximum generalized satisfiability problems
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
The hardness of approximation: gap location
Computational Complexity
Approximation algorithms for NP-hard problems
Closure properties of constraints
Journal of the ACM (JACM)
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
On the algebraic structure of combinatorial problems
Theoretical Computer Science
Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems
Journal of the ACM (JACM)
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
Some APX-completeness results for cubic graphs
Theoretical Computer Science
The complexity of maximal constraint languages
STOC '01 Proceedings of the thirty-third 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
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
The Approximability of Constraint Satisfaction Problems
SIAM Journal on Computing
The complexity of solving equations over finite groups
Information and Computation
Tractable conservative Constraint Satisfaction Problems
LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
A Short Guide To Approximation Preserving Reductions
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Learnability of quantified formulas
Theoretical Computer Science
Inapproximability results for equations over finite groups
Theoretical Computer Science - Special issue on automata, languages and programming
Classifying the Complexity of Constraints Using Finite Algebras
SIAM Journal on Computing
Supermodular functions and the complexity of MAX CSP
Discrete Applied Mathematics - Special issue: Boolean and pseudo-boolean funtions
A dichotomy theorem for constraint satisfaction problems on a 3-element set
Journal of the ACM (JACM)
The Approximability of Three-valued MAX CSP
SIAM Journal on Computing
A Simple Algorithm for Mal'tsev Constraints
SIAM Journal on Computing
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
Handbook of Constraint Programming (Foundations of Artificial Intelligence)
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights
Journal of Computer and System Sciences
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The approximability of MAX CSP with fixed-value constraints
Journal of the ACM (JACM)
Linear programming, width-1 CSPs, and robust satisfaction
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Tight bounds on the approximability of almost-satisfiable Horn SAT and exact hitting set
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Robust satisfiability of constraint satisfaction problems
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
The complexity of finite-valued CSPs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Hi-index | 5.23 |
An instance of the maximum constraint satisfaction problem (Max CSP) is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Maxk-SAT and Max Cut) and is consequently NP-hard. Thus, it is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. The PCP theorem is equivalent to the existence of a constraint language for which Max CSP has a hard gap at location 1; i.e. it is NP-hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints can be satisfied) is currently known to be NP-hard, have a certain algebraic property. We prove that any constraint language with this algebraic property makes Max CSP have a hard gap at location 1 which, in particular, implies that such problems cannot have a PTAS unless P=NP. We then apply this result to Max CSP restricted to a single constraint type; this class of problems contains, for instance, Max Cut and Max DiCut. Assuming PNP, we show that such problems do not admit PTAS except in some trivial cases. Our results hold even if the number of occurrences of each variable is bounded by a constant. Finally, we give some applications of our results.