A polynomial-time algorithm for solving NP-hard problems in practice

Constraint-based optimization, which is NP-hard, describes a large class of problems in many fields, especially in formulating early perception tasks. For example, Shape from shading of a polyhedron [12, 11] in computer vision can be formulated under such a paradigm. Some of the traditional problem-solving methods, like the exhaustive search, guarantee finding the global solution, but are too expensive in practice. Local Search, Simulated Annealing, Tabu Search, and the evolutionary algorithms have no general conditions to stop searching when the global optimum is what we are looking for. In this paper, we will present a polynomial-time algorithm [5 ]for solving the practical problems. The algorithm has many interesting computational properties not possessed by the classic ones. It is guaranteed to converge linearly, insensitive to the perturbations to its initial conditions and intermediate solutions. The closeness of the solutions to the optimal ones in cost is always provided by the algorithm so that we can stop search when the gap is closed or is small enough. It has a number of sufficient conditions for identifying optimal solutions, and a number of necessary conditions for reducing the search spaces. With 1,000 test problems randomly generated by computer, the Local Search algorithm with multiple restarts has a successful rate of only 0.2% after 25 iterations, and the Simulated Annealing algorithm [7, 2 ] with a linear cooling scheme has a successful rate of 1.5% after 250,000 steps. In contrast, the new algorithm achieved a successful rate of over 98% after only 7.27 iterations on average.