▁are ▁defined ▁by ▁choosing ▁edges ▁at ▁random ▁and ▁modifying ▁the ▁matching ▁accordingly . ▁The ▁resulting ▁Mark ov ▁chain ▁is ▁rapidly ▁mixing ▁and ▁leads ▁to ▁“ su fficient ly ▁random ” ▁match ings , ▁which ▁can ▁be ▁used ▁to ▁recover ▁the ▁partition ▁function ▁using ▁random ▁sampling . ▁The ▁resulting ▁algorithm ▁is ▁a ▁fully ▁polynomial - time ▁random ized ▁approximation ▁scheme ▁( f pr as ). ▁ ▁Comput ational ▁complexity ▁Several ▁computational ▁problems ▁are ▁associated ▁with ▁the ▁T ut te ▁polynomial . ▁The ▁most ▁straightforward ▁one ▁is ▁Input : ▁A ▁graph ▁ ▁Output : ▁The ▁coefficients ▁of ▁▁ ▁In ▁particular , ▁the ▁output ▁allows ▁evalu ating ▁ ▁which ▁is ▁equivalent ▁to ▁counting ▁the ▁number ▁of ▁ 3 - col our ings ▁of ▁G . ▁This ▁latter ▁question ▁is ▁# P - complete , ▁even ▁when ▁restricted ▁to ▁the ▁family ▁of ▁plan ar ▁graphs , ▁so ▁the ▁problem ▁of ▁computing ▁the ▁coefficients ▁of ▁the ▁T ut te ▁polynomial ▁for ▁a ▁given ▁graph ▁is ▁# P - hard ▁even ▁for ▁plan ar ▁graphs . ▁ ▁Much ▁more ▁attention ▁has ▁been ▁given ▁to ▁the ▁family ▁of ▁problems ▁called ▁T ut te ▁defined ▁for ▁every ▁complex ▁pair ▁: ▁Input : ▁A ▁graph ▁ ▁Output : ▁The ▁value ▁of ▁▁ ▁The ▁hard ness ▁of ▁these ▁problems ▁var ies ▁with ▁the ▁coordinates ▁. ▁ ▁Ex act ▁computation ▁ ▁If ▁both ▁x ▁and ▁y ▁are ▁non - negative ▁integers , ▁the ▁problem ▁ ▁belongs ▁to ▁# P . ▁For ▁general ▁integer ▁pairs , ▁the ▁T ut te ▁polynomial ▁contains ▁negative ▁terms ,