▁. ▁ ▁Another ▁restriction ▁on ▁the ▁valu ations ▁is ▁that ▁they ▁are ▁piece wise - constant ▁- ▁for ▁each ▁agent , ▁there ▁are ▁at ▁most ▁m ▁desired ▁intervals , ▁and ▁the ▁agent ' s ▁value - dens ity ▁in ▁each ▁interval ▁is ▁constant . ▁Under ▁this ▁assumption , ▁a ▁connected ▁env y - free ▁allocation ▁can ▁be ▁found ▁by ▁solving ▁▁ ▁linear ▁programs . ▁Thus , ▁when ▁n ▁is ▁constant , ▁the ▁problem ▁is ▁polynomial ▁in ▁m . ▁▁ ▁Free ▁dispos al ▁is ▁a ▁third ▁work - around ▁that ▁keeps ▁the ▁requirement ▁that ▁the ▁division ▁be ▁totally ▁env y - free ▁and ▁works ▁for ▁all ▁value ▁measures , ▁but ▁dro ps ▁the ▁requirement ▁that ▁the ▁entire ▁c ake ▁must ▁be ▁divided . ▁I . e , ▁it ▁allows ▁to ▁divide ▁a ▁subset ▁of ▁the ▁c ake ▁and ▁disc ard ▁the ▁remainder . ▁ ▁Without ▁further ▁requirements ▁the ▁problem ▁is ▁trivial , ▁since ▁it ▁is ▁always ▁env y - free ▁to ▁give ▁nothing ▁to ▁all ▁agents . ▁ ▁Thus , ▁the ▁real ▁goal ▁is ▁to ▁give ▁each ▁agent ▁a ▁strictly ▁positive ▁value . ▁Every ▁c ake ▁allocation ▁can ▁be ▁character ized ▁by ▁its ▁level ▁of ▁proportional ity , ▁ ▁which ▁is ▁the ▁value ▁of ▁the ▁least ▁fort un ate ▁agent . ▁An ▁env y - free ▁division ▁of ▁the ▁entire ▁c ake ▁is ▁also ▁a ▁proportional ▁division , ▁and ▁its ▁proportional ity ▁level ▁is ▁at ▁least ▁, ▁which ▁is ▁the ▁best ▁possible . ▁But ▁when ▁free ▁dispos al ▁is ▁allowed , ▁an ▁env y