On packing dijoins in digraphs and weighted digraphs

In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let $D=(V,A)$ be a digraph, and let $w\in \mathbb{Z}^A_{\geq 0}$. Suppose every dicut has weight at least $\tau$, for some integer $\tau\geq 2$. Let $\rho(\tau,D,w):=\frac{1}{\tau}\sum_{v\in V} m_v$, where each $m_v$ is the integer in $\{0,1,\ldots,\tau-1\}$ equal to $w(\delta^+(v))-w(\delta^-(v))$ mod $\tau$. In this paper, we prove the following results, amongst others: (1) If $w={\bf 1}$, then $A$ can be partitioned into a dijoin and a $(\tau-1)$-dijoin. (2) If $\rho(\tau,D,w)\in \{0,1\}$, then there is an equitable $w$-weighted packing of dijoins of size $\tau$. (3) If $\rho(\tau,D,w)= 2$, then there is a $w$-weighted packing of dijoins of size $\tau$. (4) If $w={\bf 1}$, $\tau=3$, and $\rho(\tau,D,w)=3$, then $A$ can be partitioned into three dijoins. Each result is best possible: (1) and (4) do not hold for general $w$, (2) does not hold for $\rho(\tau,D,w)=2$ even if $w={\bf 1}$, and (3) does not hold for $\rho(\tau,D,w)=3$. The results are rendered possible by a \emph{Decompose, Lift, and Reduce procedure}, which turns $(D,w)$ into a set of \emph{sink-regular weighted $(\tau,\tau+1)$-bipartite digraphs}, each of which is a weighted digraph where every vertex is a sink of weighted degree $\tau$ or a source of weighted degree $\tau,\tau+1$, and every dicut has weight at least $\tau$. Our results give rise to a number of approaches for resolving Woodall's Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the $\tau=2$ Conjecture for the clutter of minimal dijoins. They also show an intriguing connection to Barnette's Conjecture.Comment: 71 page

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 $\tau\geq 1$, suppose $d_A^+(U)+(\frac{\tau}{k}-1)d_A^-(U)\geq \tau$ 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 $f(U)\geq \frac{k}{\tau}(d_A^+(U)-d_A^-(U))$ 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 $\tau$-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.Comment: 29 pages, 4 figure

Woodall's conjecture and the Lucchesi-Younger theorem

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Kolja Knauer[en] This project is about directed cuts, directed joins and their packings. We study the open problem of Woodall’s Conjecture, a problem studied by many authors to such an extent that it even has a $5000 dollars prize for his demonstration set by G. Cornuéjols. We will also cover the Lucchesi-Younger Theorem proof which can be seen as a dual result of the conjecture and the counterexample for the Edmonds-Giles Conjecture, the weighted version of Woodall’s Conjecture. Besides studying the theory we have set ourselves the goal of proposing a program that given a graph, checks if all of its orientations validate the Woodall’s Conjecture. This program should be able to prove the conjecture, up to a certain number of vertices, by testing all the different combinations

The Edmonds-Giles Conjecture and its Relaxations

Given a directed graph, a directed cut is a cut with all arcs oriented in the same direction, and a directed join is a set of arcs which intersects every directed cut at least once. Edmonds and Giles conjectured for all weighted directed graphs, the minimum weight of a directed cut is equal to the maximum size of a packing of directed joins. Unfortunately, the conjecture is false; a counterexample was first given by Schrijver. However its ”dual” statement, that the minimum weight of a dijoin is equal to the maximum number of dicuts in a packing, was shown to be true by Luchessi and Younger. Various relaxations of the conjecture have been considered; Woodall’s conjecture remains open, which asks the same question for unweighted directed graphs, and Edmond- Giles conjecture was shown to be true in the special case of source-sink connected directed graphs. Following these inquries, this thesis explores different relaxations of the Edmond- Giles conjecture

Complexity of packing common bases in matroids

One of the most intriguing unsolved questions of matroid optimization is the characterization of the existence of $k$ disjoint common bases of two matroids. The significance of the problem is well-illustrated by the long list of conjectures that can be formulated as special cases, such as Woodall's conjecture on packing disjoint dijoins in a directed graph, or Rota's beautiful conjecture on rearrangements of bases. In the present paper we prove that the problem is difficult under the rank oracle model, i.e., we show that there is no algorithm which decides if the common ground set of two matroids can be partitioned into $k$ common bases by using a polynomial number of independence queries. Our complexity result holds even for the very special case when $k=2$. Through a series of reductions, we also show that the abstract problem of packing common bases in two matroids includes the NAE-SAT problem and the Perfect Even Factor problem in directed graphs. These results in turn imply that the problem is not only difficult in the independence oracle model but also includes NP-complete special cases already when $k=2$, one of the matroids is a partition matroid, while the other matroid is linear and is given by an explicit representation.Comment: 14 pages, 9 figure

Idealness and 2-resistant sets

A subset of the unit hypercube {0,1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this note, we study sets S⊆{0,1}n such that, for any subset X⊆{0,1}n of cardinality at most 2, S∪X is cube-ideal