For　a　given　graph　G　with　distinct　vertices　s　and　t,　nonnegative　integral　cost　and　delay　on　edges,　and　positive　integral　bound　C　and　D　on　cost　and　delay　respectively,　the　k　bi-constraint　path　(kBCP)　problem　is　to　compute　k　disjoint　st-paths　subject　to　C　and　D.　This　problem　is　known　to　be　NP-hard,　even　when　k=1　(Garey　and　Johnson,　Computers　and　Intractability,　1979).　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　(1+β,max{2,1+ln(1/β)})　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　immediately　and　a　factor-(1.567,1.567)　algorithm　by　slightly　modifying　the　algorithm.　Besides,　when　β=0,　the　algorithm　is　shown　to　be　with　ratio　(1,O(lnn)),　i.e.　it　is　an　algorithm　with　only　a　single　factor　ratio　O(lnn)　on　cost.　To　the　best　of　our　knowledge,　this　is　the　first　non-trivial　approximation　algorithm　that　strictly　obeys　the　delay　constraint　for　the　kBCP　problem.　©　2013,　Springer　Science+Business　Media　New　York.