Update CSE347_E2.md

pages/CSE347/Exam_reviews/CSE347_E2.md

@@ -55,8 +55,8 @@ List of known NP-hard problems:
 - Consider optimization problems whose decision problem variant is NP-hard. Unless P=NP, finding an optimal solution to these problems cannot be done in polynomial time.
 - In approximation algorithms, we make a trade-o↵: we’re willing to accept sub-optimal solutions in exchange for polynomial runtime.
 - The Approximation Ratio of our algorithm is the worst-case ratio of our solution to the optimal solution.
-- For minimization problems, this ratio is $$\text{Approximation Ratio} = \frac{\text{Our Solution}}{\text{Optimal Solution}}$$, since our solution will be larger than OPT.
-- For maximization problems, this ratio is $$\text{Approximation Ratio} = \frac{\text{Optimal Solution}}{\text{Our Solution}}$$, since our solution will be smaller than OPT.
+- For minimization problems, this ratio is $$\max_{l\in L}\left(\frac{c_A(l)}{c_{OPT}(l)}\right)$$, since our solution will be larger than OPT.
+- For maximization problems, this ratio is $$\min_{l\in L}\left(\frac{c_{OPT}(l)}{c_A(l)}\right)$$, since our solution will be smaller than OPT.
 - If given an algorithm, and you need to show it has some desired approximation ratio, there are a few approaches.
 - In recitation, we saw Max-Subset Sum. We found upper bounds on the optimal solution and showed that the given algorithm would always give a solution with value at least half of the upper bound, giving our approximation ratio of 2.
 - In lecture, you saw the Vertex Cover 2-approximation. Here, you would select any uncovered edge $(u, v)$ and add both u and v to the cover. We argued that at least one of u or v must be in the optimal cover, as the edge must be covered, so at every step we added at least one vertex from an optimal solution, and potentially one extra. So, the size of our cover could not be any larger than twice the optimal.