Given　a　directed　graph　with　non-negative　costs　and　delays　on　each　edge,　the　k　edge-disjoint　restricted　shortest　path　problem　(kRSP)　is　to　find　k　edge-disjoint　directed　paths　connecting　a　pair　of　distinct　vertices　s　and　t,　with　the　aim　of　minimizing　the　total　cost　subject　to　a　delay　constraint.　In　this　paper,　we　present　an　absolute　approximation　algorithm　via　the　LP-rounding　technique,　which　guarantees　to　find　k-1　edge-disjoint　paths　that　strictly　satisfy　the　delay　constraint　and　have　a　total　cost　no　more　than　that　of　an　optimum　solution.　The　key　observation　leading　to　our　approach　is　that　in　any　basic　optimal　solution　of　the　linear　programming　relaxation　for　kRSP,　the　underlying　graph　composed　of　edges　with　fractional　values　forms　exactly　a　cycle.　We　notably　show　that　the　cycle　always　contains　a　set　of　edges　that,　along　with　the　integral　edges　from　the　LP,　can　compose　a　desired　solution.　Lastly,　we　present　a　method　for　rounding　such　a　set　of　edges　from　this　cycle　to　eventually　obtain　the　desired　solution.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2025.