Emerging　applications　widely　use　field-programmable　gate　array(FPGA)prototypes　as　a　tool　to　verify　modern　very-large-scale　integration(VLSI)circuits,imposing　many　problems,including　routing　failure　caused　by　the　limited　number　of　connections　among　blocks　of　FPGAs　therein.Such　a　shortage　of　connections　can　be　alleviated　through　time-division　multiplexing(TDM),by　which　multiple　signals　sharing　an　identical　routing　channel　can　be　transmitted.In　this　context,the　routing　quality　dominantly　decides　the　performance　of　such　systems,proposing　the　requirement　of　minimizing　the　signal　delay　between　FPGA　pairs.This　paper　proposes　algorithms　for　the　routing　problem　in　a　multi-FPGA　system　with　TDM　support,aiming　to　minimize　the　maximum　TDM　ratio.The　algorithm　consists　of　two　major　stages:(1)A　method　is　proposed　to　set　the　weight　of　an　edge　according　to　how　many　times　it　is　shared　by　the　routing　requirements　and　consequently　to　compute　a　set　of　approximate　minimum　Steiner　trees.(2)A　ratio　assignment　method　based　on　the　edge-demand　framework　is　devised　for　assigning　ratios　to　the　edges　respecting　the　TDM　ratio　constraints.Experiments　were　conducted　against　the　public　benchmarks　to　evaluate　our　proposed　approach　as　compared　with　all　published　works,and　the　results　manifest　that　our　method　achieves　a　better　TDM　ratio　in　comparison.