The Frequency of Difficult Instances for MaxClique Approximation Algorithms

The MaxClique optimization problem cannot be approximated in polynomial-time within a ratio of n if NP 6= ZPP or within n if P 6= NP (H̊astad, 1999). We investigate the frequency with which approximation algorithms cannot achieve these ratios. Our results include the following for any ǫ > 0. 1. If NP 6= RP, then any polynomial-time approximation algorithm fails to approximate within n on a nonsparse set of instances. 2. If NP has positive p-dimension, then for some δ > 0, any 2 δ -time approximation algorithm fails to approximate within n on an exponentially dense set of instances. 3. If NP has positive p-dimension and strong pseudorandom generators exist, then for some δ > 0, any 2 δ -time approximation algorithm fails to approximate within n on an exponentially dense set of instances.