Let　G=(V,E)　be　a　given　directed　graph　in　which　every　edge　e　is　associated　with　two　nonnegative　costs:　a　weight　w(e)　and　a　length　l(e).　For　a　pair　of　specified　distinct　vertices　s,t∈V,　the　k-(edge)　disjoint　constrained　shortest　path　(kCSP)　problem　is　to　compute　k　(edge)　disjoint　paths　between　s　and　t,　such　that　the　total　length　of　the　paths　is　minimized　and　the　weight　is　bounded　by　a　given　weight　budget　W∈R0+.　The　problem　is　known　to　be　NP-hard,　even　when　k=　1　(Garey　and　Johnson　in　Computers　and　intractability,　1979).　Approximation　algorithms　with　bifactor　ratio　(1+1r,r(1+2(logr+1)r)(1+))　and　(1+1r,1+r)　have　been　developed　for　k=　2　in　Orda　and　Sprintson　(IEEE　INFOCOM,　pp.　727–738,　2004)　and　Chao　and　Hong　(IEICE　Trans　Inf　Syst　90(2):465–472,　2007),　respectively.　For　general　k,　an　approximation　algorithm　with　ratio　(1,O(lnn))　has　been　developed　for　a　weaker　version　of　kCSP,　the　k　bi-constraint　path　problem　which　is　to　compute　k　disjoint　st-paths　satisfying　a　given　length　constraint　and　a　weight　constraint　simultaneously　(Guo　et　al.　in　COCOON,　pp.　325–336,　2013).　This　paper　first　gives　an　approximation　algorithm　with　bifactor　ratio　(2,2)　for　kCSP　using　the　LP-rounding　technique.　The　algorithm　is　then　improved　by　adopting　a　more　sophisticated　method　to　round　edges.　It　is　shown　that　for　any　solution　output　by　the　improved　algorithm,　there　exists　a　real　number　0　≤　α≤　2　such　that　the　weight　and　the　length　of　the　solution　are　bounded　by　α　times　and　2　-　α　times　of　that　of　an　optimum　solution,　respectively.　The　key　observation　of　the　ratio　proof　is　to　show　that　the　fractional　edges,　in　a　basic　solution　against　the　proposed　linear　relaxation　of　kCSP,　exactly　compose　a　graph　in　which　the　degree　of　every　vertex　is　exactly　two.　At　last,　by　a　novel　enhancement　of　the　technique　in　Guo　et　al.　(COCOON,　pp.　325–336,　2013),　the　approximation　ratio　is　further　improved　to　(1,lnn).　©　2015,　Springer　Science+Business　Media　New　York.