Question 4: Recall the color coding algorithm to compute a k-length p…

Recall the color coding algorithm to compute a $k$-length path (connecting the regulators and receptors) with maximum weight. We had the following optimal substructure:

$W[v, X] = \max_{u \mid c(u) \in X \setminus \{c(v)\}} W[u, X \setminus \{c(v)\}] + w(u, v), \qquad 1 < |X| \le k$

Consider a set of proteins $T = \{T_1, \ldots, T_m\}$. You are asked to compute the path with maximum weight such that there are at least $x$ and at most $y$ proteins from $T$. To accommodate the same the DP matrix is modified as follows:

Instead of a 2-D matrix, we use a 3-D DP matrix $W$ such that $W[v, X, p]$ stores the maximum weight of a path of length $|X|$ ending at $v$ containing a vertex of each color in $X$ and exactly $p$ proteins from $T$. Which among the following is the correct optimal substructure for the algorithm?