Colors and Paths

Let $f(P,c)$ be the number of traversals in walk $P$ whose color is not $c$. Its cost is $\min_c f(P,c)$, so the answer is $\min_P\min_c f(P,c)=\min_c\min_P f(P,c)$. Fix a color $c$, assign weight 0 to its edges and 1 to other edges, and compute the shortest distance using 0-1 BFS. Minimize over all colors. Push zero-weight relaxations to the front of a deque and unit-weight relaxations to the back. Store the distance at insertion and discard stale entries. Time is $O(K(V+E))$ and space $O(V+E)$. Nonnegative weights mean removing a cycle cannot increase the chosen-color cost. Thus subtask 1 may enumerate all simple paths by DFS and count colors; a loose bound is $O(E!K)$, sufficient for $E\leq8$. The answer is between 0 and $V-1$, as are finite shortest distances, so all arithmetic fits signed 64-bit integers.

Solution written by GPT6.1