For　a　given　graph　$G$　with　distinct　vertices　$s,　\,　t$　and　a　given　delay　constraint　$D\in　R^{+}$,　the　$k$-disjoint　restricted　shortest　path　($k$RSP)　problem　of　computing　$k$-disjoint　minimum　cost　$st$-paths　with　total　delay　restrained　by　$D$,　is　known　to　be　NP-hard.　Bifactor　approximation　algorithms　have　been　developed　for　its　special　case　when　$k=2$,　while　no　approximation　algorithm　with　constant　single　factor　or　bifactor　ratio　has　been　developed　for　general　$k$.　This　paper　firstly　presents　a　$(k,　\,　(1+\epsilon)H(k))$-approximation　algorithm　for　the　$k$RSP　problem　by　extending　Orda＇s　factor-$(1.5,　\,　1.5)$　approximation　algorithm　\cite{orda2004efficient}.　Secondly,　this　paper　gives　a　novel　linear　programming　(LP)　formula　for　the　$k$RSP　problem.　Based　on　LP　rounding　technology,　this　paper　rounds　an　optimal　solution　of　this　formula　and　obtains　an　approximation　algorithm　within　a　bifactor　ratio　of　($2,　\,　2$).　To　the　best　of　our　knowledge,　it　is　the　first　approximation　algorithm　with　constant　bifactor　ratio　for　the　$k$RSP　problem.　Our　results　can　be　applied　to　serve　applications　in　networks　which　require　quality　of　service　and　robustness　simultaneously,　and　also　have　broad　applications　in　construction　of　survivable　networks　and　fault　tolerance　systems.　©　2012　IEEE.