This paper explores the use of artificial intelligence (AI) to discover novel combinatorial structures that improve the limitations of efficient algorithms. Specifically, we use the LLM coding agent AlphaEvolve to study the average-case hardness for MAX-CUT and the MAX-Independent Set, as well as the worst-case approximate hardness for MAX-k-CUT. Using AlphaEvolve, we obtain near-optimal upper and lower bounds for the authentication algorithms for MAX-CUT and the MAX-Independent Set for 3- and 4-regular graphs, and discover novel gadget reductions to obtain new non-approximate results for MAX-4-CUT and MAX-3-CUT.