The algorithm by Ferson et al. is surprisingly fast: An NP-hard optimization problem solvable in almost linear time with high probability

Abstract

We start with the algorithm of Ferson et al. (\emph{Reliable computing} {\bf 11}(3), p.~207--233, 2005), designed for solving a certain NP-hard problem motivated by robust statistics. First, we propose an efficient implementation of the algorithm and improve its complexity bound to $O(n \log n+n\cdot 2^\omega)$, where $\omega$ is the clique number in a certain intersection graph. Then we treat input data as random variables (as it is usual in statistics) and introduce a natural probabilistic data generating model. On average, we get $2^\omega = O(n^{1/\log\log n})$ and $\omega = O(\log n / \log\log n)$. This results in average computing time $O(n^{1+\epsilon})$ for $\epsilon > 0$ arbitrarily small, which may be considered as ``surprisingly good'' average time complexity for solving an NP-hard problem. Moreover, we prove the following tail bound on the distribution of computation time: ``hard'' instances, forcing the algorithm to compute in time $2^{\Omega(n)}$, occur rarely, with probability tending to zero faster than exponentially with $n \rightarrow \infty$