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Ahmad Abdi

Publications and source records attributed to Ahmad Abdi.

17 recordsLinked to original sources
arXiv2026

Testing the max-flow min-cut property and the replication conjecture

The replication conjecture [Conforti and Cornuéjols, 1993] states that every clutter with the packing property has the MFMC property. If true, this conjecture would have far-reaching consequences from integer programming and combinatorial optimization to commutative algebra. In this paper, we set out to verify the conjecture for the cuboid of a set-system in which the Hamming graph induced on the infeasible points has degree at most $δ$. The family of cuboids of degree at most $δ$ contains a rich source of clutters with the packing property, including all clutters over a ground set of size at most $δ$. We prove that any minimal counterexample must have dimension at most $δ$, thus making the target search space finite. We then use a state-of-the-art SAT solver to verify the replication conjecture for cuboids of degree at most $9$, and for clutters over at most $10$ elements. Our computational verification relies crucially on another theoretical result, that to verify the MFMC property of a clutter over $n$ elements, it suffices to check finitely many weight vectors, namely $w\in \left\{0,1,\ldots,t\right\}^n$ where $t\leq\max\{\lceil n/2\rceil, \lfloor n-\sqrt{4n+1}+1\rfloor\}$. The upper bound of $t$ improves the previous best upper bound by algebraists, which could be exponential in $n$.

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arXiv2026

Integral bases, perfect matchings, and the Petersen graph

Let $G=(V,E)$ be a matching-covered graph, denote by $P$ its perfect matching polytope, and by $L$ the integer lattice generated by the integral points in $P$. In this paper, we give short, polyhedral proofs for two difficult results established by Lovász (1987), and by Carvalho, Lucchesi, and Murty (2002) in a series of three papers totaling over 120 pages. More specifically, we prove that $L$ has a lattice basis consisting solely of incidence vectors of some perfect matchings of $G$, $2x\in L$ for all $x\in \mathrm{lin}(P)\cap \mathbb{Z}^E$, and if $G$ has no Petersen brick then $L = \mathrm{lin}(P)\cap \mathbb{Z}^E$. Our proof avoids major technical aspects of the previous proofs, the most important of these being a characterization of the dual lattice, and a `Petersen-brick-sensitive' ear decomposition result for matching-covered graphs. This is achieved by a novel study of the facial structure of the polytope $P$ and its relationship with the lattice $L$. It is also based on a first-of-its-kind polyhedral characterization of the Petersen graph.

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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

Cosigning Crossing Families and Outer-Planar Gadgets

Let $F$ be a crossing family over ground set $V$, that is, for any two sets $U,W\in{F}$ with nonempty intersection and proper union, both sets $U\cap{W},U\cup{W}$ are in $F$. Let $σ:V\to \{+,-\}$ be a signing. We call $σ$ a "cosigning" if every set includes a positive element and excludes a negative element. It is "$\cap\cup$-closed" if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively. We characterize the existence of ($\cap\cup$-closed) cosignings $σ$ through necessary and sufficient conditions. Our proofs are algorithmic and lead to elegant `forcing' algorithms for finding $σ$, reminiscent of the Cameron-Edmonds algorithm for bicoloring balanced hypergraphs. We prove that the algorithms run in polynomial time, and further, the cosigning algorithm can be run in oracle polynomial time through an application of submodular function minimization. Cosigned crossing families arise naturally in digraphs with vertex set $V$ comprised of sources and sinks, where every set in $F$ is "covered" by an incoming arc. Under mild and necessary conditions, we build an outer-planar arc covering of $F$ when the vertices are placed around a circle. These gadgets are then used to find disjoint dijoins in $0,1$-weighted planar digraphs when the weight-$1$ arcs form a connected component that is not necessarily spanning.

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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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arXiv2024

Strong orientation of a connected graph for a crossing family

Given a connected graph $G=(V,E)$ and a crossing family $\mathcal{C}$ over ground set $V$ such that $|δ_G(U)|\geq 2$ for every $U\in \mathcal{C}$, we prove there exists a strong orientation of $G$ for $\mathcal{C}$, i.e., an orientation of $G$ such that each set in $\mathcal{C}$ has at least one outgoing and at least one incoming arc. This implies the main conjecture in Chudnovsky et al. (Disjoint dijoins. Journal of Combinatorial Theory, Series B, 120:18--35, 2016). In particular, in every minimal counterexample to the Edmonds-Giles conjecture where the minimum weight of a dicut is $2$, the arcs of nonzero weight must be disconnected.

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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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arXiv2023

From coordinate subspaces over finite fields to ideal multipartite uniform clutters

Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper, we determine when the clutter $\mathcal{C}$ is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $τ=2$ Conjectures for this class of clutters.

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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

Packing odd $T$-joins with at most two terminals

Take a graph $G$, an edge subset $Σ\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,Σ,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from $Σ$. Let $ν$ be the maximum number of edge-disjoint odd $T$-joins. A signature is a set of the form $Σ\triangle δ(U)$ where $U\subseteq V(G)$ and $|U\cap T)$ is even. Let $τ$ be the minimum cardinality a $T$-cut or a signature can achieve. Then $ν\leq τ$ and we say that $(G,Σ,T)$ packs if equality holds here. We prove that $(G,Σ,T)$ packs if the signed graft is Eulerian and it excludes two special non-packing minors. Our result confirms the Cycling Conjecture for the class of clutters of odd $T$-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing $T$-joins with at most four terminals, and a new result on covering edges with cuts.

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arXiv2012

On the mixing set with a knapsack constraint

The mixing set with a knapsack constraint arises as a substructure in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution. Recently, Luedtke et al. (2010) and Küçükyavuz (2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (equivalently when the knapsack reduces to a cardinality constraint), with only minor results in the general case. In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the inequalities previously derived for the equal probabilities case. We also show that one can separate over a large class of inequalities in polynomial time.

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