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Multihoming is widely used by Internet service providers (ISPs) to obtain improved performance and reliability when connecting to the Internet. Recently, the use of overlay routing for network application traffic is rapidly increasing. As a source of both routing oscillation and cost increases, overlay routing is known to bring challenges to ISPs. In this paper, we study the interaction between overlay routing and a multihomed ISP's routing strategy with a Nash game model, and propose a routing strategy for the multihomed ISP to alleviate the negative impact of overlay traffic.We prove that with the proposed routing strategy, the network routing game can always converge to a stable state, and the ISP can reduce costs to a relatively low level. From numerical simulations, we show the efficiency and convergence resulting from the proposed routing strategy. We also discuss the conditions under which the multihomed ISP can realize minimum cost by the proposed strategy.