On the complexity of the sub-permutation problem

We study various computational aspects of the problem of determining whether a given order contains a given sub-order. Formally, given a permutation π on k elements, and a permutation σ on n > k elements, the goal is to determine whether there exists a strictly increasing function f from [1.k] to [1.n] which is order preserving, i.e., f satisfies σ(f (i)) > σ(f(j)) whenever π(i) > π(j). We call this decision problem the Sub-Permutation Problem. The study falls into two parts. In the first part we develop and analyze an algorithm (or, rather, an algorithmic paradigm) for this problem. We show that the complexity of this algorithm is at most O(n^1+C(π)), where C(π) is a naturally defined function of the permutation π. In the second part we study C(π). In particular, we show that C(π) ≤0.35k + o(k), implying that the complexity of the Sub-Permutation problem is O(c_k + n^0.35k+o(k)). On the other hand, we prove that for most π’s, C(π) = Ω(k), establishing a lower bound for our algorithm. In addition, we develop a fast polylogarithmic approximation algorithm for computing C(π), and bound the value of this parameter for some interesting families of permutations.