Packet classification has been a key technology to quickly identify an action to be taken on a packet at a switch. Several advanced applications, most of which have been introduced recently with the advent of software-defined networking, commonly require the identification of a combination of switch actions, i.e., the network-wide forwarding behavior of a packet. Conventional classification methods, however, fail to well support network-wide behaviors, since the search space is partitioned in convoluted manner due to the complexity posed by the combinations possible. This paper proposes a novel packet classification method that supports the fast determination of network-wide forwarding behaviors. To avoid the inefficiencies of existing methods, which are revealed for the first time by our research, we base our method on a compressed data structure named the multi-valued decision diagram. On the solid foundation of decision diagrams, several algorithms are introduced with thorough theoretical analyses, and the construction process and the classification performance are highly optimized for the new classification problem. Experiments on real network datasets show that our method identifies the network-wide forwarding behaviors at the line basic rate, e.g., 10 Mpps, on a single CPU core with only tens of MB of memory.