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In this paper, we present a new routing paradigm that generalizes opportunistic routing in wireless mesh networks. In multirate anypath routing, each node uses both a set of next hops and a selected transmission rate to reach a destination. Using this rate, a packet is broadcast to the nodes in the set and one of them forwards the packet on to the destination. To date, there is no theory capable of jointly optimizing both the set of next hops and the transmission rate used by each node. We bridge this gap by introducing a polynomial-time algorithm to this problem and provide the proof of its optimality. The proposed algorithm runs in the same running time as regular shortest-path algorithms and is therefore suitable for deployment in link-state routing protocols. We conducted experiments in a 802.11b testbed network, and our results show that multirate anypath routing performs on average 80% and up to 6.4 times better than anypath routing with a fixed rate of 11 Mbps. If the rate is fixed at 1 Mbps instead, performance improves by up to one order of magnitude.