We will discuss the Beigel-Eppstein 3-coloring algorithm, which is based on the symbol system problem and runs in time 1.35^n. Unlike previous algorithms, our algorithm is not based on maximal independent sets. Then we will discuss a new recursive algorithm for the maximum independent set problem. This algorithm runs in time 1.22^n and is the fastest polynomial space algorithm known for the problem (improving on results of Tarjan-Trojanowski, Jian, and Robson). Unlike previous algorithms for this problem, our algorithm may add edges as it progresses.