Random MAX SAT, random MAX CUT, and their phase transitions
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part II
Random Structures & Algorithms
Dismantling sparse random graphs
Combinatorics, Probability and Computing
Sparse graphs: Metrics and random models
Random Structures & Algorithms
Tight bounds for LDPC and LDGM codes under MAP decoding
IEEE Transactions on Information Theory
Perfect matchings as IID factors on non-amenable groups
European Journal of Combinatorics
Independent sets in random graphs from the weighted second moment method
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
The condensation transition in random hypergraph 2-coloring
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Exponential lower bounds for DPLL algorithms on satisfiable random 3-CNF formulas
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
Unsatisfiability bounds for random CSPs from an energetic interpolation method
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Going after the k-SAT threshold
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Limits of local algorithms over sparse random graphs
Proceedings of the 5th conference on Innovations in theoretical computer science
| Hi-index | 0.00 |
We establish the existence of free energy limits for several sparse random hypergraph models corresponding to certain combinatorial models on Erdos-Renyi (ER) graph G(N,c/N) and random r-regular graph G(N,r). For a variety of models, including independent sets, MAX-CUT, Coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to infinity. In the zero temperature case, this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p., thus resolving an open problem, (see Conjecture 2.20 in Wormald's Random Graph Models, or Aldous's list of open problems. Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli. Among other applications, this method was used to prove the existence of free energy limits for Viana-Bray and K-SAT models on ER graphs. The case of zero temperature was treated by taking limits of positive temperature models. We provide instead a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of ER graph G(N,c/N) and random regular graph G(N,r). In addition we establish the large deviations principle for the satisfiability property for constraint satisfaction problems such as Coloring, K-SAT and NAE-K-SAT.