1. Find pieces where every bridge needs to be used.
2. Eliminate bridges intersecting an existing bridge.
3. Eliminate bridges that make the total exceed the max.
4. Eliminate bridges that block too many other bridges.
5. Eliminate bridges that disconnect the puzzle. The simplest case is two pieces both having the number 1, but look for other possibilities.
As you eliminate or insert some bridges, recheck all strategies to look for new problems.