Abstract
Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with K=3,4, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the backtracking survey propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossible to solve in linear time when all solutions contain frozen variables.
Highlights
Discrete combinatorial optimization has a central role in many scientific disciplines, for hard problems we lack linear time algorithms that would allow us to solve very large instances
In the region aaSIDoaoas the problem is satisfiable for large N, but at present no algorithm can find solutions there. To fill this gap we study a new algorithm for finding solutions to random K-SAT problems, the backtracking survey propagation (BSP) algorithm
The probability of finding a solution increases both with r and N, but an extrapolation to the large N limit of these data is unlikely to provide a reliable estimation of the algorithmic threshold aaBSP
Summary
Discrete combinatorial optimization has a central role in many scientific disciplines, for hard problems we lack linear time algorithms that would allow us to solve very large instances. We study random K-satisfiability problems with K 1⁄4 3,4, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. The study of random K-SAT problems and of the related solving algorithms is likely to shed light on the origin of the computational complexity and to allow for the development of improved solving algorithms Both numerical[8] and analytical[9,10] evidence suggest that a threshold phenomenon takes place in random K-SAT ensembles: in the limit of very large formulas, N-N, a typical formula has a solution for aoas(K), while it is unsatisfiable for a4as(K). If we define the energy function EðxÞ as the number of unsatisfied clauses in configuration x, it has been found[12] that for a4ad the energy EðxÞ has exponentially many (in N) local minima of positive energy, which may trap algorithms that look for solutions by energy relaxation (for example, Monte Carlo simulated annealing)
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