The　min-min　problem　of　finding　a　disjoint　path　pair　with　the　length　of　the　shorter　path　minimized　is　known　to　be　NP-complete　(Xu　et　al.,　2006)　[1].　In　this　paper,　we　prove　that　in　planar　digraphs　the　edge-disjoint　min-min　problem　remains　NP-complete　and　admits　no　K-approximation　for　any　K＞1　unless　P=NP.　As　a　by-product,　we　show　that　this　problem　remains　NP-complete　even　when　all　edge　costs　are　equal　(i.e.,　stronglyNP-complete).　To　our　knowledge,　this　is　the　first　NP-completeness　proof　for　the　edge-disjoint　min-min　problem　in　planar　digraphs.　©　2012　Elsevier　B.V.　All　rights　reserved.