Let G be a directed graph on n vertices. 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 runs in
time O(2.851kβ nO(1)β logW) for graphs with integer weights in
the range [βW,W]. This is also the best currently known deterministic
algorithm for SIMPLE k-PATH- where the running time is the same without the
logW factor. We define Lkβ(n)β[n]k 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)β nO(1) for Lkβ(n) implies an algorithm of running time O(f(k)β nO(1)β 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. and its derandomization by Chen et al. for
SIMPLE k-PATH can be used to construct an acylic NFA for Lkβ(n) of size
Oβ(4k+o(k)).
We show, on the other hand, that any NFA for Lkβ(n) must be size at least
2k. We thus propose closing this gap and determining the smallest NFA for
Lkβ(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
Oβ(2k) for SIMPLE k-PATH