Problem

Aptive Environmental wants to optimize a simplified technician schedule for a single day. Given travel times between stops, implement a function that returns the minimum total travel cost to start at the Aptive branch, visit every assigned stop exactly once, and return to the branch.

Formal Specification

Write a function optimize_aptive_route(travel_times) where:

• travel_times is an n x n matrix of non-negative integers
• travel_times[i][j] is the cost to travel from stop i to stop j
• stop 0 is the Aptive branch (start and end)
• the function returns an integer: the minimum possible total route cost

If travel_times is empty, return 0. If the matrix is not square, contains negative values, or has inconsistent row lengths, raise ValueError.

Examples

Example 1 Input: travel_times = [[0,10,15,20],[10,0,35,25],[15,35,0,30],[20,25,30,0]] Output: 80 Explanation: One optimal route is 0 -> 1 -> 3 -> 2 -> 0 with cost 10 + 25 + 30 + 15 = 80.

Example 2 Input: travel_times = [[0,5],[7,0]] Output: 12 Explanation: The only valid route is 0 -> 1 -> 0.

Constraints

• 0 <= n <= 12
• travel_times.length == n
• travel_times[i].length == n
• 0 <= travel_times[i][j] <= 10^6
• Diagonal values are typically 0, but the algorithm should not rely on symmetry.