This paper explores the use of AI technology to discover novel combinatorial structures that improve existing limitations of efficient algorithms. Specifically, we use AlphaEvolve (an LLM coding agent) to study the average case hardness for MAX-CUT and MAX-Independent Sets, as well as the worst-case approximate hardness for MAX-k-CUT. Through this, we improve the upper and lower bounds of the authentication algorithm for MAX-CUT and MAX-Independent Sets, and obtain new non-approximate probability results for MAX-4-CUT and MAX-3-CUT. Furthermore, we propose a method for efficiently improving the verification process using AlphaEvolve.