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Gérard Cornuéjols

Publications and source records attributed to Gérard Cornuéjols.

At least 19 recordsLinked to original sources
arXiv2026

The Lasserre Rank of the Cropped Hypercube

In an $n$-dimensional cropped hypercube each of the $2^n$ cropping inequalities chops off a single corner of the $0$--$1$ hypercube by an $\ell_1$-distance $ρ$. The case $ρ= 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $ρ= 1/2$, $n \geq 2$, is the smallest integer $0\leq t \leq n$ such that $Δ_t < 0$ in the recurrence $Δ_{-1} = 1$, $Δ_{0} = n-1$, $Δ_t = (n-1)Δ_{t-1} - t(n-t+1)Δ_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \log^2 n)$. Asymptotically, the rank is $\frac{n}{2} + c_{1/2}\sqrt{n} + o(\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \approx 0.3825$. In fact, we prove such results for any fixed $0 < ρ< 1$.

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arXiv2026

A Hereditary Property of Cutting Plane Procedures

Let $K'$ denote the closure of a convex set $K$ under a given cutting-plane procedure. The procedure satisfies the \emph{hereditary property} if $F' = K' \cap F$ for every face $F$ of $K$. The property underlies inductive proofs of finite-rank and the polyhedrality of closures, and it is an admissibility requirement in abstract frameworks for cutting-plane procedures. Yet, it has not been studied systematically across well-known cutting-plane procedures. We establish two sufficient conditions for the property. The first applies to procedures that can be expressed as closures under intersection cuts from a family of lattice-free convex sets, when a single family realizes the closure of $K$ and of each of its faces. It yields the hereditary property for the split, lift-and-project, Lovász--Schrijver, Sherali--Adams, and Lasserre closures, applied over general convex sets and for faces that need not be exposed. The second applies to procedures whose cuts are valid inequalities for Gomory's corner polyhedron, and it yields the hereditary property for the closure of Dantzig cuts derived from all bases. We complement these results with four classical procedures for which the hereditary property fails, namely the closure of Gomory's fractional cuts (derived from either all bases or feasible bases only), the mixed-integer Chvátal closure, the $+$-cut closure, and the closure of Dantzig cuts derived from feasible bases.

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arXiv2026

A Min-Max Relation on Dicuts and Dijoins in Weighted Chordal Digraphs

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmonds-Giles conjecture is true if the underlying undirected graph is chordal. We also give a strongly polynomial-time algorithm to construct such a packing.

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arXiv2026

Lower bounds for cube-ideal set-systems

A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential lower bounds on the size of cube-ideal set-systems, and linear lower bounds on their VC dimension. We then provide applications to graph theory and combinatorial optimization, specifically to strong orientations, perfect matchings, dijoins, and ideal clutters, including the Lovász-Plummer conjecture.

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arXiv2026

Strongly connected orientations and integer lattices

Let $D=(V,A)$ be a digraph whose underlying undirected graph is $2$-edge-connected, and let $P$ be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes $D$ strongly connected. We study the lattice theoretic properties of the integer points contained in a proper face $F$ of $P$ not contained in $\{x:x_a=i\}$ for any $a\in A,i\in \{0,1\}$. We prove under a mild necessary condition that $F\cap \{0,1\}^A$ contains an integral basis $B$, i.e., $B$ is linearly independent, and any integral vector in the linear hull of $F$ is an integral linear combination of $B$. This result is surprising as the integer points in $F$ do not necessarily form a Hilbert basis. In proving the result, we develop a theory similar to Matching Theory for degree-constrained dijoins in bipartite digraphs. Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of $D$, say $τ$, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any prime number $p\geq 2$, a $p$-adic packing of dijoins of value $τ$ and of support size at most $2|A|$. We also prove that the all-ones vector belongs to the lattice generated by $F\cap \{0,1\}^A$, where $F$ is the face of $P$ satisfying $x(δ^+(U))=1$ for every dicut $δ^+(U)$ with minimum size.

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arXiv2025

Approximately Packing Dijoins via Nowhere-Zero Flows

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least $3$ disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) $k$-flows, we prove that every digraph with minimum dicut size $τ$ contains $\left\lfloor\fracτ{k}\right\rfloor$ disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in $2$-edge-connected graphs (Seymour 1981) directly leads to the existence of $\left\lfloor\fracτ{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$, which can be found in polynomial time as well. The existence of nowhere-zero circular $\frac{2p+1}{p}$-flows in $6p$-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of $\left\lfloor\frac{τp}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations.

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arXiv2023

Branch-and-Bound versus Lift-and-Project Relaxations in Combinatorial Optimization

In this paper, we consider a theoretical framework for comparing branch-and-bound with classical lift-and-project hierarchies. We simplify our analysis of streamlining the definition of branch-and-bound. We introduce "skewed $k$-trees" which give a hierarchy of relaxations that is incomparable to that of Sherali-Adams, and we show that it is much better for some instances. We also give an example where lift-and-project does very well and branch-and-bound does not. Finally, we study the set of branch-and-bound trees of height at most $k$ and effectively "squeeze" their effectiveness between two well-known lift-and-project procedures.

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arXiv2023

Arc connectivity and submodular flows in digraphs

Let $D=(V,A)$ be a digraph. For an integer $k\geq 1$, a $k$-arc-connected flip is an arc subset of $D$ such that after reversing the arcs in it the digraph becomes (strongly) $k$-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a $k$-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer $τ\geq 1$, suppose $d_A^+(U)+(\fracτ{k}-1)d_A^-(U)\geq τ$ for all $U\subsetneq V, U\neq \emptyset$, where $d_A^+(U)$ and $d_A^-(U)$ denote the number of arcs in $A$ leaving and entering $U$, respectively. Let $\mathcal{C}$ be a crossing family over ground set $V$, and let $f:\mathcal{C}\to \mathbb{Z}$ be a crossing submodular function such that $f(U)\geq \frac{k}τ(d_A^+(U)-d_A^-(U))$ for all $U\in \mathcal{C}$. Then $D$ has a $k$-arc-connected flip $J$ such that $f(U)\geq d_J^+(U)-d_J^-(U)$ for all $U\in \mathcal{C}$. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams' so-called weak orientation theorem, and proves a weaker variant of Woodall's conjecture on digraphs whose underlying undirected graph is $τ$-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

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arXiv2023

Dyadic linear programming and extensions

A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.

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arXiv2023

On packing dijoins in digraphs and weighted digraphs

Let $D=(V,A)$ be a digraph. A dicut is a cut $δ^+(U)\subseteq A$ for some nonempty proper vertex subset $U$ such that $δ^-(U)=\emptyset$, a dijoin is an arc subset that intersects every dicut at least once, and more generally a $k$-dijoin is an arc subset that intersects every dicut at least $k$ times. Our first result is that $A$ can be partitioned into a dijoin and a $(τ-1)$-dijoin where $τ$ denotes the smallest size of a dicut. Woodall conjectured the stronger statement that $A$ can be partitioned into $τ$ dijoins. Let $w\in \mathbb{Z}^A_{\geq 0}$ and suppose every dicut has weight at least $τ$, for some integer $τ\geq 2$. Let $ρ(τ,D,w):=\frac{1}τ\sum_{v\in V} m_v$, where each $m_v$ is the integer in $\{0,1,\ldots,τ-1\}$ equal to $w(δ^+(v))-w(δ^-(v))$ mod $τ$. We prove the following results: (i) If $ρ(τ,D,w)\in \{0,1\}$, then there is an equitable $w$-weighted packing of dijoins of size $τ$. (ii) If $ρ(τ,D,w)= 2$, then there is a $w$-weighted packing of dijoins of size $τ$. (iii) If $ρ(τ,D,w)=3$, $τ=3$, and $w={\bf 1}$, then $A$ can be partitioned into three dijoins. Each result is best possible: (i) does not hold for $ρ(τ,D,w)=2$ even if $w=\1$, (ii) does not hold for $ρ(τ,D,w)=3$, and (iii) do not hold for general $w$.

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arXiv2022

Total dual dyadicness and dyadic generating sets

A vector is \emph{dyadic} if each of its entries is a dyadic rational number, i.e. of the form $\frac{a}{2^k}$ for some integers $a,k$ with $k\geq 0$. A linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ for $w$ integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of \emph{dyadic generating sets for cones and subspaces}, the former being the dyadic analogue of \emph{Hilbert bases}, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the \emph{density} of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matrices, $T$-joins, cycles, and perfect matchings of a graph.

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arXiv2022

Testing idealness in the filter oracle model

A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that, given any clutter $\mathcal{C}$ over $2n$ elements via a filter oracle, decides whether or not $\mathcal{C}$ is ideal. We prove that in the worst case, $\mathfrak{A}_{2n}$ must make at least $2^n$ calls to the filter oracle. Our proof uses the theory of cuboids.

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arXiv2021

Clean tangled clutters, simplices, and projective geometries

A clutter is \emph{clean} if it has no delta or the blocker of an extended odd hole minor, and it is \emph{tangled} if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the $τ=2$ Conjecture. Let $\mathcal{C}$ be a clean tangled clutter. It was recently proved that $\mathcal{C}$ has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the {\it core} of $\mathcal{C}$. The core is a duplication of the cuboid of a set of $0-1$ points, called the {\it setcore} of $\mathcal{C}$. In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then $\mathcal{C}$ has the clutter of the lines of the Fano plane as a minor. Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

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arXiv2020

Idealness of $k$-wise intersecting families

A clutter is \emph{$k$-wise intersecting} if every $k$ members have a common element, yet no element belongs to all members. We conjecture that, for some integer $k\geq 4$, every $k$-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $k=4$ for the class of binary clutters. Two key ingredients for our proof are Jaeger's $8$-flow theorem for graphs, and Seymour's characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.

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arXiv2018

Incorporating Black-Litterman Views in Portfolio Construction when Stock Returns are a Mixture of Normals

In this paper, we consider the basic problem of portfolio construction in financial engineering, and analyze how market-based and analytical approaches can be combined to obtain efficient portfolios. As a first step in our analysis, we model the asset returns as a random variable distributed according to a mixture of normal random variables. We then discuss how to construct portfolios that minimize the Conditional Value-at-Risk (CVaR) under this probabilistic model via a convex program. We also construct a second-order cone representable approximation of the CVaR under the mixture model, and demonstrate its theoretical and empirical accuracy. Furthermore, we incorporate the market equilibrium information into this procedure through the well-known Black-Litterman approach via an inverse optimization framework by utilizing the proposed approximation. Our computational experiments on a real dataset show that this approach with an emphasis on the market equilibrium typically yields less risky portfolios than a purely market-based portfolio while producing similar returns on average.

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arXiv2016

Optimality certificates for convex minimization and Helly numbers

We consider the problem of minimizing a convex function over a subset of R^n that is not necessarily convex (minimization of a convex function over the integer points in a polytope is a special case). We define a family of duals for this problem and show that, under some natural conditions, strong duality holds for a dual problem in this family that is more restrictive than previously considered duals.

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arXiv2014

Unique Minimal Liftings for Simplicial Polytopes

For a minimal inequality derived from a maximal lattice-free simplicial polytope in $\R^n$, we investigate the region where minimal liftings are uniquely defined, and we characterize when this region covers $\R^n$. We then use this characterization to show that a minimal inequality derived from a maximal lattice-free simplex in $\R^n$ with exactly one lattice point in the relative interior of each facet has a unique minimal lifting if and only if all the vertices of the simplex are lattice points.

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