Which Algorithms Are Feasible and Which Are Not: Fuzzy Techniques Can Help in Formalizing the Notion of Feasibility

Abstract

Some algorithms are practically feasible, in the sense that for all inputs of reasonable length they provide the result in reasonable time. Other algorithms are not practically feasible, in the sense that they may work well for small-size inputs, but for slightly larger – but still reasonable-size – inputs, the computation time becomes astronomical (and not practically possible). How can we describe practical feasibility in precise terms? The usual formalization of the notion of feasibility states that an algorithm is feasible if its computation time is bounded by a polynomial of the size of the input. In most cases, this definition works well, but sometimes, it does not: e.g., according to this definition, every algorithm requiring a constant number of computational steps is feasible, even when this number of steps is larger than the number of particles in the Universe. In this paper, we show that by using fuzzy logic, we can naturally come up with a more adequate description of practical feasibility.