Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

Here we study the NP-complete $K$-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random $K$-SAT, with $\alpha=M/N$ as control parameter, can be solved quickly for small enough values of $\alpha$. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study $K$-SAT ($K=3,4$) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random $K$-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random $K$-SAT more complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure

Combinatorial approach to the interpolation method and scaling limits in sparse random graphs

We establish the existence of free energy limits for several combinatorial models on Erd\"{o}s-R\'{e}nyi graph $\mathbb {G}(N,\lfloor cN\rfloor)$ and random $r$-regular graph $\mathbb {G}(N,r)$. 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. This resolves an open problem which was proposed by Aldous (Some open problems) as one of his six favorite open problems. It was also mentioned as an open problem in several other places: Conjecture 2.20 in Wormald [In Surveys in Combinatorics, 1999 (Canterbury) (1999) 239-298 Cambridge Univ. Press]; Bollob\'{a}s and Riordan [Random Structures Algorithms 39 (2011) 1-38]; Janson and Thomason [Combin. Probab. Comput. 17 (2008) 259-264] and Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].Comment: Published in at http://dx.doi.org/10.1214/12-AOP816 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ground state of the Bethe-lattice spin glass and running time of an exact optimization algorithm

We study the Ising spin glass on random graphs with fixed connectivity z and with a Gaussian distribution of the couplings, with mean \mu and unit variance. We compute exact ground states by using a sophisticated branch-and-cut method for z=4,6 and system sizes up to N=1280 for different values of \mu. We locate the spin-glass/ferromagnet phase transition at \mu = 0.77 +/- 0.02 (z=4) and \mu = 0.56 +/- 0.02 (z=6). We also compute the energy and magnetization in the Bethe-Peierls approximation with a stochastic method, and estimate the magnitude of replica symmetry breaking corrections. Near the phase transition, we observe a sharp change of the median running time of our implementation of the algorithm, consistent with a change from a polynomial dependence on the system size, deep in the ferromagnetic phase, to slower than polynomial in the spin-glass phase.Comment: 10 pages, RevTex, 10 eps figures. Some changes in the tex

Survey-propagation decimation through distributed local computations

We discuss the implementation of two distributed solvers of the random K-SAT problem, based on some development of the recently introduced survey-propagation (SP) algorithm. The first solver, called the "SP diffusion algorithm", diffuses as dynamical information the maximum bias over the system, so that variable nodes can decide to freeze in a self-organized way, each variable making its decision on the basis of purely local information. The second solver, called the "SP reinforcement algorithm", makes use of time-dependent external forcing messages on each variable, which let the variables get completely polarized in the direction of a solution at the end of a single convergence. Both methods allow us to find a solution of the random 3-SAT problem in a range of parameters comparable with the best previously described serialized solvers. The simulated time of convergence towards a solution (if these solvers were implemented on a distributed device) grows as log(N).Comment: 18 pages, 10 figure