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

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

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

Structure Analysis of Some Generalizations of Matchings and Matroids under Algorithmic Aspects of Matchings and Matroids Under Algorithmic Aspects

Combinatorial optimization problems whose underlying structures are matchings or matroids are well-known to be solvable with efficient algorithms. Matroids can even be characterized by a simple greedy algorithm. In the first part of this thesis, some generalizations of matroids which allow the ground set to be partially ordered are considered. In particular, it will be shown that a special type of lattice polyhedra, for which Dietrich and Hoffman recently established a dual greedy algorithm, can be reduced to ordinary polymatroids. Moreover, strong exchange structures, Gauss greedoids and Delta-matroids will be extended from Boolean lattices to general distributive lattices, and the resulting structures will be characterized by certain greedy-type algorithms. While a matching of maximal size can be determined by a polynomial algorithm, the dual problem of finding a vertex cover of minimal size in general graphs is one of the hardest problems in combinatorial optimization. However, in case the graph belongs to the class of K\"onig-Egerv\'ary graphs, a maximum matching can be used to construct a minimum vertex cover. Lovasz and Korach characterized König-Egervary graphs by the exclusion of forbidden subgraphs. In the second part of this dissertation, the structure of König-Egervary graphs and the more general Red/Blue-split graphs will be analyzed. Red/Blue-split graphs have red and blue colored edges and the vertices of which can be split into two stable sets with respect to the red and blue edges, respectively. An algorithm that either determines a feasible partition of the vertices, or returns a red-blue colored subgraph (called ``flower'') characterizing non-Red/Blue-split graphs will be presented. This characterization allows the deduction of Lovasz and Korach's characterizations of König-Egerv\'ary graphs in case the red edges of the flower form a maximum matching. Furthermore, weighted Red/Blue-split graphs which model integrally solvable simple systems are introduced. A simple system is an inequality system where the sum of absolute values in each row of the integral matrix does not exceed the value two. A shortest-path algorithm and the presented Red/Blue-split algorithm will be used to find an integral solution of a simple system. These two algorithms lead to a characterization of weighted Red/Blue-split graphs by forbidden weighted subgraphs

Technique étendue d’allocation mémoire basée sur les réseaux entiers

This work extends lattice-based memory allocation, an earlier work on memory (array)reuse analysis. The main motivation is to handle in a better way the more general forms ofspecifications we see today, e.g., with loop tiling, pipelining, and other forms of parallelism availablein explicitly parallel languages. Our extension has two complementary aspects. We show howto handle more general specifications where conflicting constraints (those that describe the arrayindices that cannot share the same location) are specified as a (non-convex) union of polyhedra.Unlike convex specifications, this also requires to be able to choose suitable directions (or basis) ofarray reuse. For that, we extend two dual approaches, previously proposed for a fixed basis, intooptimization schemes to select suitable basis. Our final approach relies on a combination of thetwo, also revealing their links with, on one hand, the construction of multi-dimensional schedulesfor parallelism and tiling (but with a fundamental difference that we identify) and, on the otherhand, the construction of universal reuse vectors (UOV), which was only used so far in a specificcontext, for schedule-independent mapping.Ce travail étend l’allocation mémoire basée sur les réseaux entiersprécédemment proposée en analyse de réutilisation mémoire (de tableaux). Lamotivation principale est de traiter de meilleure façon les formes plus généralesde spécifications rencontrées aujourd’hui, comportant du tuilage de boucles,du pipeline, et d’autres formes de parallélisme exprimées dans les langages àparallélisme explicite. Notre extension a deux aspects complémentaires. Nousmontrons comment nous pouvons prendre en compte des spécifications plusgénérales où les contraintes de conflit (celles qui décrivent les indices de tableauxqui ne peuvent pas partager le même emplacement mémoire) sont spécifiées parune union (non-convexe) de polyèdres. Au contraire des spécifications convexes,ceci requiert d’être capable de choisir des directions (c’est-à-dire une base)adéquates de réutilisation des cases de tableaux. Pour cela, nous étendons deuxapproches duales, précédemment proposées pour une base fixée, en des schémasd’optimisation permettant de choisir des bases adaptées. Notre approche finaleconsiste en une combinaison des deux approches, révélant également des liensavec, d’une part, la construction d’ordonnancements multi-dimensionnels pour leparallélisme et le tuilage (avec une différence fondamentale que nous identifions)et, d’autre part, la construction de vecteurs de réutilisation universelle (UOV),qui étaient utilisés jusqu’à présent uniquement dans un contexte spécifique, celuides allocations valides pour tout ordonnancement

Detection of hidden structures on all scales in amorphous materials and complex physical systems: basic notions and applications to networks, lattice systems, and glasses

Recent decades have seen the discovery of numerous complex materials. At the root of the complexity underlying many of these materials lies a large number of possible contending atomic- and larger-scale configurations and the intricate correlations between their constituents. For a detailed understanding, there is a need for tools that enable the detection of pertinent structures on all spatial and temporal scales. Towards this end, we suggest a new method by invoking ideas from network analysis and information theory. Our method efficiently identifies basic unit cells and topological defects in systems with low disorder and may analyze general amorphous structures to identify candidate natural structures where a clear definition of order is lacking. This general unbiased detection of physical structure does not require a guess as to which of the system properties should be deemed as important and may constitute a natural point of departure for further analysis. The method applies to both static and dynamic systems.Comment: (23 pages, 9 figures