Finding the shortest path to visit all non-blocked squares on a grid

Question

Let's say you have a grid like this (made randomly):

Now let's say you have a car starting randomly from one of the while boxes, what would be the shortest path to go through each one of the white boxes? You can visit each white box as many times as you want and can't jump over the black boxes. The black boxes are like walls. In simple words you can move from white box to white box only.

You can move in any direction, even diagonally.

Two subquestions:

1. Assume you know the position of all black boxes before moving.
2. Assume you only know the position of a black box when you are in a white box adjacent to it.

Answer 1

You should model the problem as a complete graph where the distance between two nodes (white boxes) is the length of the shortest path between those nodes. Those path lengths can be calculated by the Floyd-Warshall algorithm. Then, you can treat it as "Traveling salesman", like glowcoder wrote.

EDIT: to make it more clear: you can describe each "interesting" path through the maze by a sequence of different white boxes. Because if you have an arbitrary path visiting each white box, you can split it up into sub-paths each one ending at a new white box not visited so far. Each of this sub-paths from white box A to B can be replaced by a shortest sub-path from A to B, that's why you need the shortest-paths-between-all-pairs-of-nodes matrix.