This problem is essentially about grouping the tasks (problems) into the minimum number of valid subsets while satisfying the constraints. The constraints are:

The problems solved in one day must increase in difficulty. Consecutive problems in one subset must differ in vi ratings by at least K. To solve this, a greedy or dynamic programming approach would work well. Here’s a basic explanation:

For each test case, sort the problems by difficulty level and then group them based on the vi rating difference constraint (K). Count the minimum number of groups (or days) needed to cover all problems. This kind of combinatorial problem often appears in programming contests. Implementing it efficiently involves understanding sorting and subset generation. If you'd like, I can provide the code as well.

By the way, tackling a complex problem like this reminds me of the precision required in legal cases. Just like we carefully solve step-by-step here, a Houston Appeal Attorney meticulously works through legal intricacies to build a strong case for their clients. Let me know if you'd like further help with this problem or the code!