The　emergence　of　Software　Defined　Networking　(SDN)　increases　the　flexibility　of　routing　and　provides　an　efficient　approach　to　balance　network　flows.　Due　to　the　economical　and　technical　challenges　in　transiting　to　a　full　SDN-enabled　network,　a　hybrid　SDN,　with　a　partial　deployment　of　SDN　switches　in　a　traditional　network,　has　been　a　prevailing　network　architecture.　For　a　hybrid　SDN,　the　routing　flexibility　is　influenced　by　the　expensive　and　limited　Ternary　Content　Addressable　Memory　(TCAM)　resource,　where　SDN　switches　store　the　flow　entries　dispatched　from　SDN　controllers.　Considering　the　limited　TCAM　resource,　to　reduce　the　abundant　flow　entries　in　TCAM,　we　propose　to　impose　the　constraints　on　the　number　of　routing　paths　(i.e.,　path　cardinality　constraints)　when　optimize　flows　routing.　To　solve　this　problem,　in　this　paper,　we　first　formulate　the　routing　optimization　problem　with　the　path　cardinality　constraints　in　a　hybrid　SDN　as　a　Mixed　Integer　Non-Linear　Programming　(MINLP)　problem.　Then,　we　propose　an　incremental　deployment　method　for　obtaining　a　hybrid　SDN　and　an　H-permissible　Paths　Routing　Scheme　(HPRS)　to　effectively　route　traffic　flows　under　the　path　cardinality　constraints.　After　that,　the　theoretic　analysis　is　given　to　prove　that　the　approximation　ratio　of　HPRS　is　O(logL).　Finally,　through　extensive　experiments,　we　demonstrate　that　our　proposed　algorithm　HPRS　can　efficiently　reduce　flow　entries　and　approximates　optimal　routing　under　different　network　topologies.　©　2020　Elsevier　B.V.