Graph Coloring ( 1994 , 1998 ; Karger , Motwani , Sudan )

An independent set in an undirected graph G = (V,E) is a set of vertices that induce a subgraph which does not contain any edges. The size of the maximum independent set in G is denoted by α(G). For an integer k, a k-coloring of G is a function σ : V → [1 . . . k] which assigns colors to the vertices of G. A valid k-coloring of G is a coloring in which each color class is an independent set. The chromatic number χ(G) of G is the smallest k for which there exists a valid k-coloring of G. Finding χ(G) is a fundamental NP-hard problem. Hence, when limited to polynomial time algorithms, one turns to the question of estimating the value of χ(G) or to the closely related problem of approximate coloring.