1 research outputs found
Relations between automata and the simple k-path problem
Let be a directed graph on vertices. Given an integer , the
SIMPLE -PATH problem asks whether there exists a simple -path in . In
case is weighted, the MIN-WT SIMPLE -PATH problem asks for a simple
-path in of minimal weight. The fastest currently known deterministic
algorithm for MIN-WT SIMPLE -PATH by Fomin, Lokshtanov and Saurabh runs in
time for graphs with integer weights in
the range . This is also the best currently known deterministic
algorithm for SIMPLE k-PATH- where the running time is the same without the
factor. We define to be the set of words of
length whose symbols are all distinct. We show that an explicit
construction of a non-deterministic automaton (NFA) of size for implies an algorithm of running time for MIN-WT SIMPLE -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 -PATH can be used to construct an acylic NFA for of size
.
We show, on the other hand, that any NFA for must be size at least
. We thus propose closing this gap and determining the smallest NFA for
as an interesting open problem that might lead to faster algorithms
for MIN-WT SIMPLE -PATH.
We use a relation between SIMPLE -PATH and non-deterministic xor automata
(NXA) to give another direction for a deterministic algorithm with running time
for SIMPLE -PATH