Let　G　=　(V,　E)　be　a　given　(directed)　graph　in　which　every　edge　is　with　a　cost　and　a　delay　that　are　nonnegative.　The　k-disjoint　restricted　shortest　path　(kRSP)　problem　is　to　compute　k　(edge)　disjoint　minimum　cost　paths　between　two　distinct　vertices　s,　t　∈　V,　such　that　the　total　delay　of　these　paths　are　bounded　by　a　given　delay　constraint　D　∈　ℝ0+.　This　problem　is　known　to　be　NP-hard,　even　when　k　=　1　[4].　Approximation　algorithms　with　bifactor　ratio　(1　+　1/r,　r(1+2(log　r+1)/r)(1+∈))　and　(1+1/r,　r(1+2(log　r+1)/r))　have　been　developed　for　its　special　case　when　k　=　2　respectively　in　[11]　and　[3].　For　general　k,　an　approximation　algorithm　with　ratio　(1,　O(ln　n))　has　been　developed　for　a　weaker　version　of　kRSP,　the　k　bi-constraint　path　problem　of　computing　k　disjoint　st-paths　to　satisfy　the　given　cost　constraint　and　delay　constraint　simultaneously　[7].　In　this　paper,　an　approximation　algorithm　with　bifactor　ratio　(2,　2)　is　first　given　for　the　kRSP　problem.　Then　it　is　improved　such　that　for　any　resulted　solution,　there　exists　a　real　number　0　≤　α　≤　2　that　the　delay　and　the　cost　of　the　solution　is　bounded,　respectively,　by　α　times　and　2　-　α　times　of　that　of　an　optimal　solution.　These　two　algorithms　are　both　based　on　rounding　a　basic　optimal　solution　of　a　LP　formula,　which　is　a　relaxation　of　an　integral　linear　programming　(ILP)　formula　for　the　kRSP　problem.　The　key　observation　of　the　two　ratio　proofs　is　to　show　that,　the　fractional　edges　of　a　basic　solution　to　the　LP　formula　will　compose　a　graph　in　which　the　degree　of　every　vertex　is　exactly　2.　To　the　best　of　our　knowledge,　it　is　the　first　algorithm　with　a　single　factor　polylogarithmic　ratio　for　the　kRSP　problem.　©　2014　Springer　International　Publishing.