1. A recovery path designing circuit for designing recovery paths in a multipoint communication network including a plurality of nodes and links that connect the nodes, which employs a pre-establishing-type failure-detected end switching system in which alternate paths for respective primary paths are pre-established and a node that has detected a failure switches the route from a primary path to a recovery path, comprising: an optimization reference generating means for setting an objective function to minimize a link cost; a recovery path forming condition generating means for generating a constraint expression to form a recovery path from a switching start node to an Egress node; a link capacity calculating condition generating means for generating a constraint expression to calculate a link capacity required of the respective links; a link accommodating condition generating means for generating a constraint expression to accommodate the link capacity within a range of a given link capacity; a forbidden link and node setting condition generating means for generating constraint expressions to set at least one unavailable forbidden node and link with respect to each recovery path; and an optimizing means for solving integer programming problems expressed by the objective function set by the optimization reference generating means and constraint expressions generated by the recovery path forming condition generating means, the link capacity calculating condition generating means, the link accommodating condition generating means, and the forbidden link and node setting condition generating means.