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NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proved to be easy. We show the intractability of random instances of a graph colouring problem: this graph problem is hard on average unless all NP problems under all samplable (i.e. generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities. This poses significant technical difficulties.
We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings—that we also develop in terms of Turing degrees of unsolvability.
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