Free Bits, PCPs, and Nonapproximability---Towards Tight Results
SIAM Journal on Computing
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
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
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
The Inapproximability of Lattice and Coding Problems with Preprocessing
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Approximation Algorithms for Unique Games
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Hardness of Approximating the Closest Vector Problem with Pre-Processing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Witnesses for non-satisfiability of dense random 3CNF formulas
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The PCP theorem by gap amplification
Journal of the ACM (JACM)
Unique games on expanding constraint graphs are easy: extended abstract
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Public-key cryptography from different assumptions
Proceedings of the forty-second ACM symposium on Theory of computing
Spectral Algorithms for Unique Games
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
2log1-ε n hardness for the closest vector problem with preprocessing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
The hardness of decoding linear codes with preprocessing
IEEE Transactions on Information Theory
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The factor graph of an instance of a symmetric constraint satisfaction problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND, k-LIN) is a bipartite graph describing which variables appear in which constraints. The factor graph describes the instance up to the polarity of the variables, and hence there are up to 2km instances of the CSP that share the same factor graph. It is well known that factor graphs with certain structural properties make the underlying CSP easier to either solve exactly (e.g., for tree structures) or approximately (e.g., for planar structures). We are interested in the following question: is there a factor graph for which if one can solve every instance of the CSP with this particular factor graph, then one can solve every instance of the CSP regardless of the factor graph (and similarly, for approximation)? We call such a factor graph universal. As one needs different factor graphs for different values of n and m, this gives rise to the notion of a family of universal factor graphs. We initiate a systematic study of universal factor graphs, and present some results for max-kSAT. Our work has connections with the notion of preprocessing as previously studied for closest codeword and closest lattice-vector problems, with proofs for the PCP theorem, and with tests for the long code. Many questions remain open.