Littérature scientifique sur le sujet « Box-Totally dual integral polyhedron »

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which the minimum is taken over the set of integral elements of a box total dual integral polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.

This paper contributes to the polyhedral aspect of the Maximum-Weight Bounded-Degree Rooted Tree Problem, where only edge-indexed variables are considered. An initial formulation is given, followed by an analysis of the dimension and a facial study for the polytope . Several families of new valid inequalities are proposed, which enables us to characterize the polytope on trees and cycles with a totally dual integral system.

AbstractLet $$D=(V,A)$$ D = ( V , A ) be a digraph. For an integer $$k\ge 1$$ k ≥ 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 $$\tau \ge 1$$ τ ≥ 1 , suppose $$d_A^+(U)+(\frac{\tau }{k}-1)d_A^-(U)\ge \tau $$ d A + ( U ) + ( τ k - 1 ) d A - ( U ) ≥ τ for all $$U\subsetneq V, U\ne \emptyset $$ U ⊊ V , U ≠ ∅ , where $$d_A^+(U)$$ d A + ( U ) and $$d_A^-(U)$$ d A - ( U ) denote the number of arcs in A leaving and entering U, respectively. Let $${\mathcal {C}}$$ C be a crossing family over ground set V, and let $$f:{\mathcal {C}}\rightarrow {\mathbb {Z}}$$ f : C → Z be a crossing submodular function such that $$f(U)\ge \frac{k}{\tau }(d_A^+(U)-d_A^-(U))$$ f ( U ) ≥ k τ ( d A + ( U ) - d A - ( U ) ) for all $$U\in {\mathcal {C}}$$ U ∈ C . Then D has a k-arc-connected flip J such that $$f(U)\ge d_J^+(U)-d_J^-(U)$$ f ( U ) ≥ d J + ( U ) - d J - ( U ) for all $$U\in {\mathcal {C}}$$ U ∈ 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 $$\tau $$ 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.

Dans cette thèse, nous étudions les polyèdres total dual box-intègraux (box-TDI) associés à plusieurs problèmes et matrices totalement équimodulaires. De plus, nous étudions la complexité de certaines questions fondamentales liées à ces polyèdres. Nous commençons par considérer les matrices totalement équimodulaires, qui sont des matrices telles que, pour chaque sous-ensemble de lignes linéairement indépendantes, toutes les sous-matrices maximales non-singulières ont le même déterminant en valeur absolue. Malgré leurs similitudes avec les matrices totalement unimodulaires, nous mettons en évidence plusieurs différences, même dans le cas des matrices d'incidence et d'adjacence des graphes. Comme on le sait, la matrice d'incidence d'un graphe donné est totalement unimodulaire si et seulement si le graphe est biparti. Cependant, la totale équimodularité d'une matrice d'incidence dépend du fait que nous considérons la représentation sommet-arête ou arête-sommet. Nous fournissons des caractérisations pour les deux cas. En conséquence, nous prouvons que reconnaître si un polyèdre donné est box-TDI est un problème co-NP-complet. La caractérisation de la totale unimodularité ou de la totale équimodularité de la matrice d'adjacence d'un graphe biparti donné reste non résolue, alors que nous avons résolu le problème correspondant dans le cas de la totale équimodularité lorsque le graphe est non-biparti. Dans une dernière partie, nous caractérisons les graphes pour lesquels le polytope des couplages parfaits (PMP) est décrit par des inégalités triviales et des inégalités correspondant à des coupes serrées. Les coupes serrées sont définies comme des coupes qui partagent précisément une arête avec chaque couplage parfait. Nous prouvons ensuite que tout graphe pour lequel le PMP correspondant est box-TDI appartient à cette classe. En conséquence, reconnaître si le PMP est box-TDI est un problème résoluble en temps polynomial. Cependant, nous fournissons plusieurs contre-exemples montrant que cette classe de graphes ne garantit pas la box-TDIness du PMP. Enfin, nous présentons des conditions nécessaires pour un polytope de couverture des arêtes pour être box-TDI et caractérisons quand le polytope des couplages extensibles est box-TDI, qui est l'enveloppe convex des couplages inclus dans un couplage parfait<br>In this thesis, we study box-totally dual integral (box-TDI) polyhedra associated with severalproblems and totally equimodular matrices. Moreover, we study the complexity of some funda-mental questions related to them.We start by considering totally equimodular matrices, which are matrices such that, forevery subset of linearly independent rows, all nonsingular maximal submatrices have the samedeterminant in absolute value. Despite their similarities with totally unimodular matrices, wehighlight several differences, even in the case of incidence and adjacency matrices of graphs.As is well-known, the incidence matrix of a given graph is totally unimodular if and only if thegraph is bipartite. However, the total equimodularity of an incidence matrix depends on whetherwe consider the vertex-edge or the edge-vertex representation. We provide characterizations forboth cases. As a consequence, we prove that recognizing whether a given polyhedron is box-TDIis a co-NP-complete problem.Characterizing the total unimodularity or total equimodularity of the adjacency matrix of agiven bipartite graph remains unsolved, while we solved the corresponding problem in the case oftotal equimodularity when the graph is nonbipartite.In a later part of this work, we characterize the graphs for which the perfect matching polytope(PMP) is described by trivial inequalities and the inequalities corresponding to tight cuts. Tightcuts are defined as cuts that share precisely one edge with each perfect matching. We thenprove that any graph for which the corresponding PMP is box-TDI belongs to this class. Asa consequence, it turns out that recognizing whether the PMP is box-TDI is a polynomial-timeproblem. However, we provide several counterexamples showing that this class of graphs does notguarantee the box-TDIness of the PMP.Lastly, we present necessary conditions for the box-TDIness of the edge cover polytope andcharacterize the box-TDIness of the extendable matching polytope, which is the convex hull ofthe matchings included in a perfect matching