Note on Pseudolattices, Lattices and Submodular Linear Programs

A pseudolattice L is a poset with lattice-type binary operations. Assuming that the pseudolattice permits a modular representation as a family of subsets of a set U with certain compatibility properties, we show that L actually is a distributive lattice with the same supremum operation. Given a submodular function r:L o R , we prove that the corresponding unrestricted linear program relative to the representing set family can be solved by a greedy algorithm. This complements the Monge algorithm of Dietrich and Hoffman for the associated dual linear program. We furthermore show that our Monge and greedy algorithm is generally optimal for nonnegative submodular linear programs and their duals (relative to L )

An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in ${\mathbb R}^n$. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u

Lattice polyhedra and submodular flows

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice

Note on Representations of Ordered Semirings

The article studies ordered semigroups and semirings with respect to their representations in lattices. Such structures are essentially the pseudolattices of Dietrich and Hoffman. It is shown that a subadditive representation implies the semigroup to be a lattice in its own right. In particular, distributive lattices can be characterized as semirings admitting subadditive supermodular representations. The cover problem asks for a minimal cover of a ground set by representing sets with respect to a semiring. A greedy algorithm is exhibited to solve the cover problem for the class of lattices with weakly subadditive and supermodular representation

K-submodular functions and convexity of their Lovász extension

AbstractWe consider a class of lattice polyhedra introduced by Hoffman and Schwartz. The polyhedra are defined in terms of a kind of submodular function defined on the set of antichains of a poset. Recently, Krüger (Discrete Appl. Math. 99 (2000) 125–148) showed the validity of a greedy algorithm for this class of lattice polyhedra, which had been proved by Faigle and Kern to be valid for a less general class of polyhedra. In this paper, we investigate submodular functions in Krüger's sense and associated polyhedra. We show that the Lovász extension of a submodular function in Krüger's sense is convex, and vice versa. Furthermore, we show a polynomial-time algorithm to test whether or not a vector is an extreme point of the associated polyhedron

Cooperative Games with Lattice Structure

A general model for cooperative games with possibly restricted and hierarchically ordered coalitions is introduced and shown to have lattice structure under quite general assumptions. Moreover, the core of games with lattice structure is investigated. Within a general framework that includes the model of classical cooperative games as a special case, it is proved algorithmically that monotone convex games have a non-empty core. Finally, the solution concept of the Shapley value is extended to the general class of cooperative games with restricted cooperation. It is shown that several generalizations of the Shapley value that have been proposed in the literature are subsumed in this model

Structural Aspects Of Ordered Polymatroids

This paper generalizes some aspects of polymatroid theory to partially ordered sets. The investigations are mainly based on Faigle and Kern, Submodular Linear Programs on Forests, Mathematical Programming 72 (1996). A slightly modified concept of submodularity is introduced. As a consequence the main results do not require any assumptions concerning the underlying partially ordered groundset of the polymatroid. The partial orders are not required to be rooted forests. We consider two different basis concepts for ordered polymatroids. These are Core(f), the set of all elements with maximal cardinality and Max(f), the set of all maximal feasible elements. Both concepts are equivalent for unordered polymatroids. The sets Core(f) and Max(f) are completely described by facet-inducing inequalities. Furthermore it is shown by an example that Max(f) is in general not a polyhedral set