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

Greedy algorithms and poset matroids

We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes

Exploring the On-line Partitioning of Posets Problem

One question relating to partially ordered sets (posets) is that of partitioning or dividing the poset\u27s elements into the fewest number of chains that span the poset. In 1950, Dilworth established that the width of the poset - the size of the largest set composed only of incomparable elements - is the minimum number of chains needed to partition that poset. Such a bound in on-line partitioning has been harder to establish, and work has evalutated classes of posets based on their width. This paper reviews the theorems that established val(2)=5 and illustrates them with examples. It also covers some of the work on establishing bounds for on-line partitioning with the Greedy Algorithm. The paper concludes by contributing a bound on incomparable elements in graded, (t+t)-free, finite width posets

The Knapsack Problem with Neighbour Constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.Comment: Full version of IWOCA 2011 pape

An algorithmic characterization of antimatroids

In an article entitled “Optimal sequencing of a single machine subject to precedence constraints” E.L. Lawler presented a now classical minmax result for job scheduling. In essence, Lawler's proof demonstrated that the properties of partially ordered sets were sufficient to solve the posed scheduling problem. These properties are, in fact, common to a more general class of combinatorial structures known as antimatroids, which have recently received considerable attention in the literature. It is demonstrated that the properties of antimatroids are not only sufficient but necessary to solve the scheduling problem posed by Lawler, thus yielding an algorithmic characterization of antimatroids. Examples of problems solvable by the general result are provided

Computing Possible and Certain Answers over Order-Incomplete Data

This paper studies the complexity of query evaluation for databases whose relations are partially ordered; the problem commonly arises when combining or transforming ordered data from multiple sources. We focus on queries in a useful fragment of SQL, namely positive relational algebra with aggregates, whose bag semantics we extend to the partially ordered setting. Our semantics leads to the study of two main computational problems: the possibility and certainty of query answers. We show that these problems are respectively NP-complete and coNP-complete, but identify tractable cases depending on the query operators or input partial orders. We further introduce a duplicate elimination operator and study its effect on the complexity results.Comment: 55 pages, 56 references. Extended journal version of arXiv:1707.07222. Up to the stylesheet, page/environment numbering, and possible minor publisher-induced changes, this is the exact content of the journal paper that will appear in Theoretical Computer Scienc

On the Complexity of Mining Itemsets from the Crowd Using Taxonomies

We study the problem of frequent itemset mining in domains where data is not recorded in a conventional database but only exists in human knowledge. We provide examples of such scenarios, and present a crowdsourcing model for them. The model uses the crowd as an oracle to find out whether an itemset is frequent or not, and relies on a known taxonomy of the item domain to guide the search for frequent itemsets. In the spirit of data mining with oracles, we analyze the complexity of this problem in terms of (i) crowd complexity, that measures the number of crowd questions required to identify the frequent itemsets; and (ii) computational complexity, that measures the computational effort required to choose the questions. We provide lower and upper complexity bounds in terms of the size and structure of the input taxonomy, as well as the size of a concise description of the output itemsets. We also provide constructive algorithms that achieve the upper bounds, and consider more efficient variants for practical situations.Comment: 18 pages, 2 figures. To be published to ICDT'13. Added missing acknowledgemen

A Fast Algorithm for Well-Spaced Points and Approximate Delaunay Graphs

We present a new algorithm that produces a well-spaced superset of points conforming to a given input set in any dimension with guaranteed optimal output size. We also provide an approximate Delaunay graph on the output points. Our algorithm runs in expected time $O(2^{O(d)}(n\log n + m))$, where $n$ is the input size, $m$ is the output point set size, and $d$ is the ambient dimension. The constants only depend on the desired element quality bounds. To gain this new efficiency, the algorithm approximately maintains the Voronoi diagram of the current set of points by storing a superset of the Delaunay neighbors of each point. By retaining quality of the Voronoi diagram and avoiding the storage of the full Voronoi diagram, a simple exponential dependence on $d$ is obtained in the running time. Thus, if one only wants the approximate neighbors structure of a refined Delaunay mesh conforming to a set of input points, the algorithm will return a size $2^{O(d)}m$ graph in $2^{O(d)}(n\log n + m)$ expected time. If $m$ is superlinear in $n$, then we can produce a hierarchically well-spaced superset of size $2^{O(d)}n$ in $2^{O(d)}n\log n$ expected time.Comment: Full versio