Real-world problems such as ad allocation and matching havebeen extensively studied under the lens of combinatorial optimization. In several applications, uncertainty in the input appears naturally and this has led to the study of online stochastic optimization models for such problems.

Real-world problems such as ad allocation and matching have been extensively studied under the lens of combinatorial optimization. In several applications, uncertainty in the input appears naturally and this has led to the study of online stochastic optimization models for such problems. For the offline case, these constrained combinatorial optimization problems have been extensively studied, and Contention Resolution Schemes (CRSs), introduced by Chekuri, Vondrák, and Zenklusen, have emerged in recent years as a general framework to obtaining a solution. The idea behind a CRS is to first obtain a fractional solution to a (continuous) relaxation of the objective and then round the fractional solution to an integral one. When the order of rounding is controlled by an adversary, Online Contention Resolution Schemes (OCRSs) can be used instead, and have been successfully applied in settings such as prophet inequalities and stochastic probing. In this work, we focus on greedy OCRSs, which provide guarantees against the strongest possible adversary, an almighty adversary. Intuitively, a greedy OCRS has to make all its decisions before the online process starts.

The reduction of covering decision systems is an important problem in data mining, and covering-based rough sets serve as an efficient technique to process the problem. Geometric lattices have been widely used in many fields, especially greedy algorithm design which plays an important role in the reduction problems. Therefore, it is meaningful to combine coverings with geometric lattices to solve the optimization problems. In this paper, we obtain geometric lattices from coverings through matroids and then apply them to the issue of attribute reduction. First, a geometric lattice structure of a covering is constructed through transversal matroids. Then its atoms are studied and used to describe the lattice. Second, considering that all the closed sets of a finite matroid form a geometric lattice, we propose a dependence space through matroids and study the attribute reduction issues of the space, which realizes the application of geometric lattices to attribute reduction. Furthermore, a special type of information system is taken as an example to illustrate the application. In a word, this work points out an interesting view, namely, geometric lattice to study the attribute reduction issues of information systems.

Covering-based rough set theory is a useful tool to deal with inexact, uncertain or vague knowledge in information systems. Geometric lattice has widely used in diverse fields, especially search algorithm design which plays important role in covering reductions. In this paper, we construct four geometric lattice structures of covering-based rough sets through matroids, and compare their relationships. First, a geometric lattice structure of covering-based rough sets is established through the transversal matroid induced by the covering, and its characteristics including atoms, modular elements and modular pairs are studied. We also construct a one-to-one correspondence between this type of geometric lattices and transversal matroids in the context of covering-based rough sets. Second, sufficient and necessary conditions for three types of covering upper approximation operators to be closure operators of matroids are presented. We exhibit three types of matroids through closure axioms, and then obtain three geometric lattice structures of covering-based rough sets. Third, these four geometric lattice structures are compared. Some core concepts such as reducible elements in covering-based rough sets are investigated with geometric lattices. In a word, this work points out an interesting view, namely geometric lattice, to study covering-based rough sets.