On the tractability of coloring semirandom graphs

As part of the efforts put in understanding the intricacies of the k-colorability problem, different distributions over k-colorable graphs were analyzed. While the problem is notoriously hard (not even reasonably approximable) in the worst case, the average case (with respect to such distributions) often turns out to be “easy”. Semi-random models mediate between these two extremes and are more suitable to imitate “real-life” instances than purely random models. In this work we consider semi-random variants of the planted k-colorability distribution. This continues a line of research pursued by Coja-Oghlan [7] and by Krivelevich and Vilenchik [20]. Our aim is to study a more general semi-random framework than suggested there. On the one hand we show that the algorithmic techniques developed in [20] extend to our more general semi-random setting; on the other hand we give a hardness result, proving that a closely related semi-random model is intractable. Thus, we provide some indication about which properties of the input distribution make the k-colorability problem hard. keywords: graph algorithms, average case analysis, semi-random models, k-coloring