Combinatorica
Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
The scaling window of the 2-SAT transition
Random Structures & Algorithms
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Two-coloring random hypergraphs
Random Structures & Algorithms
A Treshold for Unsatisfiability
MFCS '92 Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
Improved algorithmic versions of the Lovász Local Lemma
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Mick gets some (the odds are on his side) (satisfiability)
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Quantum SAT for a Qutrit-Cinquit Pair Is QMA1-Complete
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
The Lovász Local Lemma and Satisfiability
Efficient Algorithms
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
Proceedings of the forty-second ACM symposium on Theory of computing
Complexity of Stoquastic Frustration-Free Hamiltonians
SIAM Journal on Computing
An algorithmic approach to the lovász local lemma. I
Random Structures & Algorithms
A parallel algorithmic version of the local lemma
Random Structures & Algorithms
Proceedings of the forty-third annual ACM symposium on Theory of computing
On perfect completeness for QMA
Quantum Information & Computation
The local lemma is tight for SAT
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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The Lovász Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of “weakly dependent” criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-qsat problem (quantum analog of k-sat): For instance we show that any collection of rank-1 projectors, with the property that each qubit appears in at most 2k/(e ċ k) of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-qsat. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [2009, 2010] we greatly extend the known satisfiable region for random k-qsat to a density of Ω(2k/k2). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.