Given　a　radius　R∈Z+　and　a　set　X　of　n　points　distributed　within　a　metric　space,　we　consider　the　radius-constrained　k-median　problem,　which　combines　both　the　k-center　and　k-median　clustering　problems.　In　this　problem,　the　objective　is　the　same　as　that　of　the　k-median　problem,　with　the　additional　constraint　that　every　point　x　in　X　must　be　assigned　to　a　center　within　the　given　radius　R.　This　paper　proposes　an　approximation　algorithm　that　achieves　a　bicriteria　approximation　ratio　of　(3+ε,7)　by　incorporating　local　search　with　a　key　ball　structure.　The　algorithm　constructs　a　keyball　center　set　to　ensure　coverage　of　the　points　and　iteratively　refines　the　solution　through　subset　swaps　while　satisfying　feasibility　conditions.　Thus,　this　process　maintains　coverage　while　reducing　costs.　Compared　to　the　state-of-the-art　approximation　ratio　of　(8, 4)　based　on　a　linear　programming　formulation　for　this　problem,　our　approach　improves　the　k-median　ratio　from　8　to　3+ε,　at　the　cost　of　increasing　the　radius　ratio　from　4　to 7.　©　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature　2025.