Deploying multiple supply voltages (multi-Vdds) on one chip is an important technique to reduce dynamic power consumption. In this work we present an optimality study for resource binding targeting designs with multi-Vdds. This is similar to the voltage-island design concept, except that the granularity of our voltage island is on the functional-unit level as opposed to the core level. We are interested in achieving the maximum number of low-Vdd operations and, in the same time, minimizing switching activity during functional unit binding. To the best of our knowledge, there is no known optimal solution to this problem. To compute an optimal solution for this problem and examine the quality gap between our solution and previous heuristic solutions, we formulate this problem as a min-cost network flow problem, but with special equal-flow constraints. This formulation leads to an easy reduction to the integer linear programming (ILP) solution and also enables efficient approximate solution by Lagrangian relaxation. Experimental results show that the optimal solution computed based on our formulation provides 7% more low-Vdd operations and also reduces the total switching activity by 20% compared to one of the best known heuristic algorithms that consider multi-Vdd assignments only.