We　consider　a　new　problem　of　designing　a　network　with　small　s-t　effective　resistance.　In　this　problem,　we　are　given　an　undirected　graph　G　=　(V,　E),　two　designated　vertices　s,　t　is　an　element　of　V,　and　a　budget　k.　The　goal　is　to　choose　a　subgraph　of　G　with　at　most　k　edges　to　minimize　the　s-t　effective　resistance.　This　problem　is　an　interpolation　between　the　shortest　path　problem　and　the　minimum　cost　flow　problem　and　has　applications　in　electrical　network　design.　We　present　several　algorithmic　and　hardness　results　for　this　problem　and　its　variants.　On　the　hardness　side,　we　show　that　the　problem　is　NP-hard,　and　the　weighted　version　is　hard　to　approximate　within　a　factor　smaller　than　two　assuming　the　small-set　expansion　conjecture.　On　the　algorithmic　side,　we　analyze　a　convex　programming　relaxation　of　the　problem　and　design　a　constant　factor　approximation　algorithm.　The　key　of　the　rounding　algorithm　is　a　randomized　path-rounding　procedure　based　on　the　optimality　conditions　and　a　flow　decomposition　of　the　fractional　solution.　We　also　use　dynamic　programming　to　obtain　a　fully　polynomial　time　approximation　scheme　when　the　input　graph　is　a　series-parallel　graph,　with　better　approximation　ratio　than　the　integrality　gap　of　the　convex　program　for　these　graphs.