For　a　given　graph　G　with　positive　integral　cost　and　delay　on　edges,　distinct　vertices　s　and　t,　cost　bound　C　aˆˆˆ　Z　+　and　delay　bound　D　aˆˆˆ　Z　+,　the　k　bi-constraint　path　(kBCP)　problem　is　to　compute　k　disjoint　st-paths　subject　to　C　and　D.　This　problem　is　known　NP-hard,　even　when　k　=　1　[4].　This　paper　first　gives　a　simple　approximation　algorithm　with　factor-(2,2),　i.e.　the　algorithm　computes　a　solution　with　delay　and　cost　bounded　by　2　D　and　2　C　respectively.　Later,　a　novel　improved　approximation　algorithm　with　ratio　is　developed　by　constructing　interesting　auxiliary　graphs　and　employing　the　cycle　cancellation　method.　As　a　consequence,　we　can　obtain　a　factor-(1.369,　2)　approximation　algorithm　by　setting　and　a　factor-(1.567,　1.567)　algorithm　by　setting.　Besides,　by　setting　β　=　0,　an　approximation　algorithm　with　ratio　(1,　O(ln　n)),　i.e.　an　algorithm　with　only　a　single　factor　ratio　O(ln　n)　on　cost,　can　be　immediately　obtained.　To　the　best　of　our　knowledge,　this　is　the　first　non-trivial　approximation　algorithm　for　the　kBCP　problem　that　strictly　obeys　the　delay　constraint.　©　2013　Springer-Verlag　Berlin　Heidelberg.