I know there are algorithms for finding a point inside a simple polygon. Given a set of polygons inside a rectangle (think a bunch of polygons on a computer screen), is there an efficient algorithm for finding a point that is inside the rectangle but not inside any of the specified polygons? (Note that these polygons don't overlap, but may share a common border (or part of a border).)
If the polygons can overlap, the problem can be solved in $O(n^2)$ time (where $n$ is the number of sides of the polygons in total) by constructing the arrangement of line segments and maintaining as you construct it the number of polygons that cover each cell of the arrangement. There are $O(n^2)$ cells, arrangements can be constructed in $O(n^2)$ time, and it takes constant time per cell to maintain the covering number because it differs by one from the number of any neighboring cell.
You are unlikely to solve the problem significantly faster than $O(n^2)$ because it is equivalent in difficulty to 3SUM — see the original paper on 3SUM hardness, "On a class of $O(n^2)$ problems in computational geometry", by Gajentaan and Overmars (CGTA 1995).
With your specification that the polygons cannot overlap, the same approach works in $O(n\log n)$ time using an output-sensitive arrangement construction algorithm.