Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. IV. The General Unimodular Two-Dimensional Case

Abstract

We study an abstract setting for cutting planes for integer programming called the infinite group problem. In this abstraction, cutting planes are computed via cut generating function that act on the simplex tableau. In this function space, cut generating functions are classified as minimal, extreme, and facets as a proxy for understanding the strength or potential importance of these functions. Prior work developed algorithms for testing minimality, extremality, and facetness for cut generating functions applied to 1-row tableau and to some 2-row tableau in a restricted setting. We complement and generalize this work by giving an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem. Along the way, we develop results of independent interest on functional equations and infinite systems of linear equations.

Figures (4)

• Figure 1: The hierarchy of valid, minimal, and extreme functions and facets and weak facets. (a) General case. (b) Situation in the finite-dimensional case. As a consequence of this work, we can show some cases of 1-row and 2-row continuous piecewise linear functions with rational breakpoints where this correspondence also holds.
• Figure 2: This function is minimal, but not extreme (and hence also not a facet). We can see this as for any distinct minimal $\pi^1 = \pi + \bar{\pi}$ (blue), $\pi^2 = \pi - \bar{\pi}$ (red) such that $\pi = \tfrac{1}{2}\pi^1 + \tfrac{1}{2} \pi^2$, the functions $\pi^1$ and $\pi^2$ are continuous piecewise linear with the same breakpoints as $\pi$. A finite-dimensional extremality test can be used to find the two linearly independent perturbations $\bar{\pi}$ (magenta), as shown in basu-hildebrand-koeppe:equivariant.
• Figure 3: A minimal valid, continuous, piecewise linear function over the polyhedral complex $\mathcal{P}_5$. Left, the three-dimensional plot of the function on $D = [0,1]^2$. Right, the complex $\mathcal{P}_5$, restricted to $D$ and colored according to slopes to match the 3-dimensional plot, and decorated with values $v$ at each vertex of $\mathcal{P}_5$ where the function takes value $\tfrac{v}{4}$.
• Figure 4: The plot features sets $I_1$, $J_1$ as pink line segments on x-axis (2,0)--(3,0) and y-axis (0,3)--(0,4), respectively. $I_2$ and $J_2$ are blue line segments from (0,1)--(1,1) and (2,2)--(1,3) respectively. The pink and blue polygons represent the Minkowski sums $I_1 + J_1$ and $I_2 + J_2$. Lastly, $K_1$ and $K_2$ denote sides of set $K$, outlined by the intersections of the pink polygon and the blue polygon.

Theorems & Definitions (93)

• Theorem 2.1: Gomory and Johnson infinite
• Definition 2.1
• Theorem 2.2: MR0479416basu-hildebrand-koeppe:equivariant
• Theorem 2.3: Facet Theorem tspace, bhkm
• Definition 2.3
• Example 2.3: Breakpoint intervals in $\mathbb R^1$ basu-hildebrand-koeppe:equivariant
• Example 2.3: Standard triangulations of $\mathbb R^2$ bhk-IPCOext
• Definition 2.3
• Remark 2.3: igp_surveyigp_survey_part_2
• Definition 2.3: Valid 7-tuple