Relations between automata and the simple k-path problem

Let G be a directed graph on n vertices. A k-path in G is a path p = v1 → . . . → vk in G. Given an integer k ≤ n, the SIMPLE k-PATH problem asks whether there exists a simple k-path in G. In case G is weighted, the MIN-WT SIMPLE k-PATH problem asks for a simple k-path in G of minimal weight. The fastest currently known deterministic algorithm for MIN-WT SIMPLE k-PATH by Fomin, Lokshtanov and Saurabh [5] runs in time O(2.851 · n · logW ) for graphs with integer weights in the range [−W,W ]. This is also the best currently known deterministic algorithm for SIMPLE k-PATHwhere the running time is the same without the logW factor. We define Lk(n) ⊆ [n] to be the set of words of length k whose symbols are all distinct. We show that an explicit construction of a non-deterministic automaton (NFA) of size f(k) ·n for Lk(n) implies an algorithm of running time O(f(k) ·n · logW ) for MIN-WT SIMPLE k-PATH when the weights are non-negative or the constructed NFA is acyclic as a directed graph. We show that the algorithm of Kneis et al. [9] and its derandomization by Chen et al.[8] for SIMPLE k-PATH can be used to construct an acylic NFA for Lk(n) of size O (4). We show, on the other hand, that any NFA for Lk(n) must have size at least 2 . We thus propose closing this gap and determining the smallest NFA for Lk(n) as an interesting open problem that might lead to faster algorithms for MIN-WT SIMPLE k-PATH. We use a relation between SIMPLE k-PATH and non-deterministic xor automata (NXA) to give another direction for a deterministic algorithm with running time O(2) for SIMPLE k-PATH.