![]() One of the difficulties lies with the fact that most optimization algorithms are discrete and not easily cast in forms amenable to chaos theory methods. Complex behavior by deterministic dynamical systems is coined chaos in the literature 9, 10, 11, 12, 13, and thus the behavior of algorithms for hard problems is expected to appear highly irregular or chaotic 14.Īlthough the theory of nonlinear dynamical systems and chaos is well-established, it has not yet been exploited in the context of optimization algorithms. Thus, we expect that the dynamics of those algorithms that exploit the structure of hard problems will reflect the complexity inherent in the problem itself. Here we will only deal with deterministic algorithms that is, once an initial state is given, the “trajectory” of the dynamical system is uniquely determined. ![]() Accordingly, now 3-SAT can be solved by a deterministic algorithm with an upper bound of O(1.473 N) steps 8. To improve performance, algorithms have become more sophisticated by exploiting the structure of the problem (of the state space). For example, the simplest algorithm for the Ising model ground state problem, or the 3-SAT problem would be exhaustively testing potentially all the 2 N configurations, which quickly becomes forbidding with increasing N. An algorithm is a finite set of instructions acting in some state space, applied iteratively from an initial state until an end state is reached. In the following we treat algorithms as dynamical systems. Just as for the spin glass model, here we also have exponentially many (2 N) configurations or assignments to search. In k-SAT we are given N boolean variables to which we need to assign 1s or 0s (TRUE or FALSE) such that a given set of clauses in conjunctive normal form, each containing k or fewer literals (literal: a boolean variable or its negation) are all satisfied, i.e., evaluate to TRUE. ![]() Namely, any problem in NP can be solved via a small number of calls to a k-SAT solver and a polynomial number of steps (in the size of the input) outside the subroutine invoking the k-SAT solver. Completeness means that all problems in NP (hence Sudoku as well), can be translated in polynomial time and formulated as a k-SAT problem, as shown for the first time by Cook and Levin 2. ![]() There has been considerable work in this direction, especially for the boolean satisfiability problem SAT (or k-SAT), which is NP-complete for k ≥ 3. Since there is little hope in providing polynomial time algorithms for NP-complete problems, the focus shifted towards understanding the nature of the complexity forbidding fast solutions to these problems. Barahona 6, then Istrail 7 have shown that for non-planar crystalline lattices, the ground-state problem and computing the partition function are NP-complete 7. Additionally, to describe the statistical behavior of such Ising spin models, one has to compute the partition function, which is a sum over all the 2 N configurations. In the latter case, for the ground-state problem of Ising spin glasses (☑ spins), one needs to find the lowest energy configuration among all the 2 N possible spin configurations, where N is the number of spins. The intractability of NP-complete problems has important consequences, ranging from public-key cryptography to statistical mechanics. NP-complete problems are “intractable” (unless P = NP) 2, 5 in the sense that all known algorithms that compute solutions to them do so in exponential worst-case time (in the number of variables N) in spite of the fact that if given a candidate solution, it takes only polynomial time to check its correctness. Sudoku is an exact cover type constraint satisfaction problem 2 and it is one of Karp's 21 NP-complete problems 3, when generalized to N × N grids 4. ![]() In Sudoku, considered as one of the world's most popular puzzles 1, we have to fill in the cells of a 9 × 9 grid with integers 1 to 9 such that in all rows, all columns and in nine 3 × 3 blocks every digit appears exactly once, while respecting a set of previously given digits in some of the cells (the so-called clues). ![]()
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