Algorithms That Still Produce a Solution (Maybe Not Optimal) Even When Interrupted: Shary's Idea Justified

Many problems of interval computations are NP-hard; this means, crudely speaking, that every algorithm that finds an exact solution will, in some cases, require exponential time, which is often unrealistic. If it turns out that an algorithm works too long, then we have to stop it. Some algorithms, when stopped, produce a reasonable approximation to the solution (for example, an enclosure for the desired interval). Some other algorithms only produce a solution at the very end, so when interrupted, they produce no solution at all.For interval computations, this distinction was first described by Shary, who proposed to require that algorithms still produce a solution, maybe not optimal, even when interrupted. In order for an algorithm to still produce a solution, the algorithm has to do some extra work. The natural question is: how will it affect the computation time?In this paper, we show that even with this "extra work" requirement, computation time stays (asymptotically) the same: linear-time algorithms can be converted into linear-time interruptible ones, quadratic-time into quadratic-time interruptible, etc. Thus, from the theoretical viewpoint, Shary's idea is quite feasible.