On the effect of randomness on planted 3-coloring models.

We present the hosted coloring framework for studying al- gorithmic and hardness results for the k-coloring problem. There is a class H of host graphs. One selects a graph H ∈ H and plants in it a balanced k-coloring (by partitioning the vertex set into k roughly equal parts, and removing all edges within each part). The resulting graph G is given as input to a polynomial time algorithm that needs to k-color G (any legal k-coloring would do – the algorithm is not required to recover the planted k-coloring). Earlier planted models correspond to the case that H is the class of all n-vertex d-regular graphs, a member H ∈ H is chosen at random, and then a balanced k-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when d = n δ (for 0 < δ ≤ 1), and Alon and Kahale [1997] managed to do so even when d is a sufficiently large constant. The new aspect in our framework is that it need not in- volve randomness. In one model within the framework (with k = 3) H is a d regular spectral expander (meaning that ex- cept for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than d) chosen by an adversary, and the planted 3-coloring is ran- dom. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In an- other model H is a random d-regular graph but the planted balanced 3-coloring is chosen by an adversary, after seeing H. We show that for a certain range of average degrees somewhat below √ n, finding a 3-coloring is NP-hard. To- gether these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.