An Average Case NP-Complete Graph Coloring Problem

TL;DR

This paper demonstrates that the average-case complexity of a specific graph coloring problem is NP-complete, indicating it is hard on typical random instances unless all NP problems are easy on all samplable distributions.

Contribution

It introduces a novel reduction technique that maintains near-randomness and low distortion, establishing average-case NP-completeness for the problem.

Findings

Random instances are hard on average unless all NP problems are easy on samplable distributions.

The reduction preserves near-randomness and avoids super-polynomial probability distortion.

The problem's average-case complexity aligns with worst-case NP-completeness under the new reduction.

Abstract

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 proven easy. We show the intractability of random instances of a graph coloring problem: this graph problem is hard on average unless all NP problem 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.