2018/07/24 by Matthias Köppe, Yuan Zhou, Köppe, Matthias +1
Computer Science · Mathematics · #90C10 #Advanced Graph Theory Research #Combinatorics #Complexity and Algorithms in Graphs #Corollary #Discrete mathematics #FOS: Mathematics #Finite Group Theory Research #Function (biology) #Group (periodic table) #Group Theory (math.GR) #Infinite group #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Physics #Piecewise #Piecewise linear function #Polyhedron #Pure mathematics #Rational function #math.GR #math.OC #msc:90C10
paper · pdf · doi:10.48550/arxiv.1807.09758
21 pages, 5 figures. Dedicated to Professor Ellis L. Johnson on the occasion of his eightieth birthday. v2 corrects a mistake in the subadditivity lemma (Lemma 4.4)
openalex publication_date 2018/07/24 · arxiv created 2019/04/17 · arxiv updated 2019/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/(Mq) 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.