Discrete Applied Mathematics

An integral-valued set function f:2^v ↦ Z is called polymatroid if it is submodular, nondecreasing, and f(φ) = 0. Given a polymatroid function f and an integer threshold t ≥ 1, let α = α(f,t) denote the number of maximal sets X ⊆ V satisfying f(X) < t, ...

We consider a class of functions satisfying the gross-substitutes property (GS-functions). We show that GS-functions are concave functions, whose parquets are constituted by quasi-polymatroids. The class of conjugate functions to GS-functions turns out ...

In this paper, some problems that arise in the realization of finite state machines (FSM) are shown to be strongly related to the theory of submodular functions. Specifically, we use the idea of the principal lattice of partitions of a submodular ...

Recently, the first combinatorial strongly polynomial algorithms for submodular function minimization have been devised independently by Iwata, Fleischer, and Fujishige and by Schrijver. In this paper, we improve the running time of Schrijver's ...

Submodular functions have appeared to be a key tool for proving the monotonicity of several graph searching games. In this paper, we provide a general game theoretic framework able to unify old and new monotonicity results in a unique min-max theorem. ...

Given a simple bipartite graph G and an integer t ≥ 2, we derive a formula for the maximum number of edges in a subgraph H of G so that H contains no node of degree larger than t and H contains no complete bipartite graph K_t,t as a subgraph. In the ...

We give a constructive characterization of undirected graphs which contain k spanning trees after adding any new edge. This is a generalization of a theorem of Henneberg and Laman who gave the characterization for k = 2. We also give a constructive ...

By applying the matroid partition theorem of J. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69 (1965) 67) to a hypergraphic generalization of graphic matroids, due to Lorea (Cahiers Centre Etudes Rech. Oper. 17 (1975) 289), we obtain a generalization ...

Graph orientation is a well-studied area of combinatorial optimization, one that provides a link between directed and undirected graphs. An important class of questions that arise in this area concerns orientations with connectivity requirements. In ...

Two important branches of graph connectivity problems are connectivity augmentation, which consists of augmenting a graph by adding new edges so as to meet a specified target connectivity, and connectivity orientation, where the goal is to find an ...

A graph G = (V,E) is called minimally (k,T)-edge-connected with respect to some T ⊆ V if there exist k-edge-disjoint paths between every pair u,v ∈ T but this property fails by deleting any edge of G. We show that |V| can be bounded by a (linear) ...

We discuss matroid-likeness of polyhedra whose facets have non-01-vectors as their normal vectors. We propose, as a generalized class of submodular polyhedra, the class of down-monotone polyhedra whose support functions satisfy submodularity on non-...

Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle-Kern (Math. Programming 72 (1996) 195-206)...

We introduce two classes of discrete quasiconvex functions, called quasi M- and L-convex functions, by generalizing the concepts of M- and L-convexity due to Murota (Adv. Math. 124 (1996) 272) and (Math. Programming 83 (1998) 313). We investigate the ...

The concept of M-convex functions plays a central role in "discrete convex analysis", a unified framework of discrete optimization recently developed by Murota and others. This paper gives two new characterizations of M- and M'-convex functions ...

In this paper we relate the minimization problems for general submodular functions and symmetric submodular functions. We characterize the contractions and restrictions of symmetric submodular functions. The latter we show to be the same as posimodular ...

An antimatroid is an accessible union-closed family of subsets of a finite set. A number of classes of antimatroids are closed under taking minors such as point-search antimatroids of rooted (di)graphs, line-search antimatroids of rooted (di)graphs, ...

We investigate into the role of submodular functions in designing new heuristics and approximate algorithms to some NP-hard problems arising in the field of VLSI Design Automation. In particular, we design and implement efficient heuristic for improving ...

We prove that there is no degree of connectivity which will guarantee that a hypergraph contains two edge-disjoint spanning connected subhypergraphs. We also show that Edmonds' theorem on arc-disjoint branchings cannot be extended to directed ...