Well out of reach : Why hard problems are hard ?

We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to connrm uncolorabil-ity quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at the phase transition. Also, our results suggest that a conjectured double phase transition in graph coloring occurs only in small graphs. Similar results are likely in other NP-complete problems where instances from phase transitions are hard for all known algorithms, and will help to explain the phenomenon. Furthermore, our results help to elucidate the distinction between polynomial and non-polynomial search behavior. an EPSRC visiting fellowship, GR/M54605. We thank members of APES at Leeds and Strathclyde for their help with this work, especially Isla Ross and Toby Walsh.