, ▁the ▁Jones ▁polynomial ▁is ▁# P - hard ▁to ▁compute . ▁Finally , ▁computing ▁the ▁number ▁of ▁four - color ings ▁of ▁a ▁plan ar ▁graph ▁is ▁# P - complete , ▁even ▁though ▁the ▁decision ▁problem ▁is ▁trivial ▁by ▁the ▁four ▁color ▁theorem . ▁In ▁contrast , ▁it ▁is ▁easy ▁to ▁see ▁that ▁counting ▁the ▁number ▁of ▁three - color ings ▁for ▁plan ar ▁graphs ▁is ▁# P - complete ▁because ▁the ▁decision ▁problem ▁is ▁known ▁to ▁be ▁N P - complete ▁via ▁a ▁par sim on ious ▁reduction . ▁ ▁Appro xim ation ▁The ▁question ▁which ▁points ▁admit ▁a ▁good ▁approximation ▁algorithm ▁has ▁been ▁very ▁well ▁studied . ▁A part ▁from ▁the ▁points ▁that ▁can ▁be ▁computed ▁exactly ▁in ▁polynomial ▁time , ▁the ▁only ▁approximation ▁algorithm ▁known ▁for ▁ ▁is ▁J err um ▁and ▁S inc la ir ’ s ▁F PR AS , ▁which ▁works ▁for ▁points ▁on ▁the ▁“ I sing ” ▁hyper bol a ▁ ▁for ▁y ▁> ▁ 0 . ▁If ▁the ▁input ▁graphs ▁are ▁restricted ▁to ▁dense ▁instances , ▁with ▁degree ▁, ▁there ▁is ▁an ▁F PR AS ▁if ▁x ▁ ≥ ▁ 1 , ▁y ▁ ≥ ▁ 1 . ▁ ▁Even ▁though ▁the ▁situation ▁is ▁not ▁as ▁well ▁understood ▁as ▁for ▁exact ▁computation , ▁large ▁areas ▁of ▁the ▁plane ▁are ▁known ▁to ▁be ▁hard ▁to ▁approximate . ▁ ▁See ▁also ▁▁ ▁Bol lob ás – R i ord an ▁polynomial ▁ ▁A ▁T ut te – G ro th end ieck ▁invariant ▁is