Extreme Functions with an Arbitrary Number of Slopes

Abstract

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number k ≥ 2, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.

Equivariant perturbation in Gomory and Johnson’s infinite group problem. VI. The curious case of two-sided discontinuous minimal valid functions

“…Then θ is affine with the same slope over int(p 1 (F )), int(p 2 (F )), and int(p 3 (F )). 2 In the situation of this result, we say that the intervals int(p 1 (F )), int(p 2 (F )), and int(p 3 (F )) are (directly) covered 3 and are in the same connected covered component 4 . Subintervals of covered intervals are covered.…”

“…In this way, our paper contributes to the foundations of the cutting-plane theory of integer programming by investigating the fine structure of a space of cutgenerating functions. In this regard, the paper is in a line of recent papers on the Gomory-Johnson model: the MPA 2012 paper [1], in which the first non-piecewise linear, measurable extreme function with 2 slopes was discovered; and the IPCO 2016 paper [2], in which a measurable extreme function with an infinite number of slopes was constructed using techniques similar to those in [1].…”

Equivariant perturbation in Gomory and Johnson's infinite group problem (V). Software for the continuous and discontinuous 1-row case

“…Since the publication of [4,20], additional extreme functions have been implemented in the compendium. They include the family of extreme functions with an arbitrary number of slopes (extreme_ functions.bcdsp_arbitrary_slope) that was recently constructed by Basu-Conforti-Di Summa-Paat [8], the family of CPL = 3 functions (mlr_cpl3_...) that was obtained by Miller-Li-Richard [25] from an approximate lifting of superadditive functions, and new parametric families and sporadic extreme functions (kzh_...) that were found by Köppe-Zhou [22,23] using computer-based search.…”

“…In particular, any sub-interval of a covered interval is also covered. 8 The spaceΠ π (R, Z) of effective perturbation functions, should not be confounded withΠ E (R, Z), the space of perturbation functions with prescribed additivities E, defined in [4] as…”

Software for Cut-Generating Functions in the Gomory–Johnson Model and Beyond

“…See also our function number_of_slopes. We refer the reader to [4, section 2.4] for a discussion of the number of slopes of extreme functions, and[2] and bcdsp_ arbitrary_slope for the latest developments in this direction.…”