We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a common generalization of these problems, called symbol-system satisfiability or, briefly, SSS [R. Floyd & R. Beigel, The Language of Machines]. 3-SAT is equivalent to (2,3)-SSS while the other problems ... more >>>

We study the approximate-coloring(q,Q) problem: Given a graph G, decide whether \chi(G) \le q or \chi(G)\ge Q. We derive conditional hardness for this problem for any constant 3\le q < Q. For q \ge 4, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For q=3, we base our hardness ... more >>>