Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (2012) considered a strengthened version of Chvatal-Gomory (CG) inequalities that use 0-1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also consider mixed-integer sets defined by linear constraints, to which we extend our polyhedrality result by defining the strengthened CG inequalities on integer variables.
In this paper, we show that the Chvatal-Gomory closure of a compact convex set is a rational polytope. This resolves an open question discussed in Schrijver [Schrijver 80'] and generalizes the same result for the case of rational polytopes [Schrijver 80'], rational ellipsoids [Dey-Vielma 10'] and strictly convex sets [Dadush-Dey-Vielma 10']. In particular, it shows that the CG closure of an irrational polytope is a rational polytope, which was the open question in [Schrijver 80'].
In this paper, we provide an equivalent condition for the Chvatal-Gomory (CG) closure of a closed convex set to be finitely-generated. Using this result, we are able to prove that, for any closed convex set that can be written as the Minkowski sum of a compact convex set and a closed convex cone, its CG closure is a rational polyhedron if and only if its recession cone is a rational polyhedral cone. As a consequence, this generalizes and unifies all the currently known results, for the case of rational polyhedron and compact convex set.
Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs
Georgia Institute of Technology（佐治亚理工学院）；Hong Kong University of Science and Technology（香港科技大学）；University of Florida（佛罗里达大学）；Jacobs Technion-Cornell Institute, Cornell Tech（雅各布斯科技锡安学院，康奈尔科技）；Universidad de Chile（智利大学）
Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chvátal-Gomory closure operator have to be performed on Q to obtain a polyhedron contained in the affine hull of P. We show that if P contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on n. On the other hand, we prove that if P is not full-dimensional and does not contain any integer point in its relative interior, then no finite boun…
Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by `iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists in solving a nonlinear optimization problem o…
Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/Sum-of-Squares SOS hierarchy fails to capture. In this paper we present a novel polynomial time SOS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low-degree polynomials). We show that the new SOS hierarchy recovers the Bienstock-Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock-Zuckerberg hierarchy as a polynomial sized, efficiently constructive extended formula…
Let $S \subseteq \{0,1\}^n$ and $R$ be any polytope contained in $[0,1]^n$ with $R \cap \{0,1\}^n = S$. We prove that $R$ has bounded Chvátal-Gomory rank (CG-rank) provided that $S$ has bounded notch and bounded gap, where the notch is the minimum integer $p$ such that all $p$-dimensional faces of the $0/1$-cube have a nonempty intersection with $S$, and the gap is a measure of the size of the facet coefficients of $\mathsf{conv}(S)$. Let $H[\bar{S}]$ denote the subgraph of the $n$-cube induced by the vertices not in $S$. We prove that if $H[\bar{S}]$ does not contain a subdivision of a large …
In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper…
Split rank of a rational polyhedron is finite. The well known proof of this is based on the fact that split closure is stronger than the Chvátal closure, and the Chvátal rank of a rational polyhedron is finite due to the result of Chvátal and Schrijver. In this note we provide an independent proof for the fact that every rational polyhedron has finite split rank. In principal, we construct a nonnegative potential function which decreases by at least one with "every" second split closure unless the integer hull of the polyhedron is reached.
We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic struct…
A \emph{general branch-and-bound tree} is a branch-and-bound tree which is allowed to use general disjunctions of the form $π^{\top} x \leq π_0 \,\vee\, π^{\top}x \geq π_0 + 1$, where $π$ is an integer vector and $π_0$ is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to th…
We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-cuts) that defines the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program min{cx : x \in S \cap Z^n }, where S \subset R^n is a compact set and c \in Z^n . We analyze the number …
The $\{0,\frac{1}{2}\}$-closure of a rational polyhedron $\{ x \colon Ax \le b \}$ is obtained by adding all Gomory-Chvátal cuts that can be derived from the linear system $Ax \le b$ using multipliers in $\{0,\frac{1}{2}\}$. We show that deciding whether the $\{0,\frac{1}{2}\}$-closure coincides with the integer hull is strongly NP-hard. A direct consequence of our proof is that, testing whether the linear description of the $\{0,\frac{1}{2}\}$-closure derived from $Ax \le b$ is totally dual integral, is strongly NP-hard.
In this paper, we study the strength of Chvatal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-appro…
