We give a variant of Basu–Hildebrand–Molinaro’s approximation theorem for continuous minimal valid functions for Gomory–Johnson’s infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in 1/q Z for some q ∈ N) and that take rational values at the breakpoints. In contrast to Basu et al.’s construction, our construction preserves all function values on 1/q Z. As a corollary, we obtain that every extreme function for the finite group problem on 1/q Z is the restriction of a continuous piecewise linear two-slope extreme function for the infinite group problem with breakpoints on a refinement 1/(M q) Z for some M ∈ N. In combination with Gomory’s master theorem [Some Polyhedra related to Combinatorial Problems, Lin. Alg. Appl. 2 (1969), 451–558], this shows that the infinite group problem is the correct master problem for facets (extreme functions) of 1-row group relaxations.

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