Combinatorial optimization arises naturally in a variety of resource management problems in public transportation companies, industrial manufacturing, financial and health-care institutions. Standing out are two categories of optimization problems, packing and covering, with applications in material and manpower planning, scheduling, routing, investment and resource allocation. The sad fact is that these problems are not only NP-hard, but also provably hard to approximate: for instance, no algorithm can approximate set-cover within a logarithmic multiplicative factor or independent set within a polynomial multiplicative factor. Even worse, with the proliferation of ubiquitous data-collecting devices and fast distributed data-sharing capabilities, the inputs to these problems are so large and complex that traditional computational ideas are infeasible. The task of finding efficient, reliable (provably good) and flexible solutions to packing and covering problems remains a challenge with high potential impact.
One key to avoiding the inherent NP-hardness of these combinatorial problems is to take into account the geometric structure of data that exists in many applications. Such structures, however, become very large with data increases (which, e.g., for business data doubles worldwide every 1.2 years). This dismal state of affairs in theory then translates current applications still using heuristics with the hope (and no guarantees) that they would run fast and produce good-quality solutions. Recent effective approaches to large computational problems therefore rely on structures which are provably small; hence the field of streaming algorithms computes small `sketches' of data, or sub-linear algorithms that even avoid looking at the entire data using structures called epsilon-nets. The core of both these approaches is to show that the structural essence of the entire data can be captured by a small-sized subset. Closer to our theme, the key to the resolution of a long-standing open problem on minimum set-cover for disks was another small-sized structure, namely geometric separators.
The goal of this project is the design of provably efficient and reliable algorithms for packing and covering problems through the study of small-sized structural approximations of geometric data. This involves the analysis of geometric data (e.g., epsilon-nets) to construct small-sized geometric structures (e.g., separators) to design efficient algorithms for packing and covering problems for reliable software solution. A successful completion of this proposal would entail: new structural understanding of geometric data; leading directly to better algorithms for problems on geometric data; careful implementation and algorithmic fine-tuning; solving instances of specific problems of relevance to industry; and finally integration into the state-of-the-art libraries and technologies.
ENET: a near-linear time algorithm for epsilon-nets for disks in the plane.
DNET: algorithms for computing hitting-sets for disks in the plane.
Publications (HAL pour les publications sous SAGA)
All related publications need to mention the support from the project as follows: Supported by the French ANR jeune chercheur grant ANR SAGA JCJC-14-CE25-0016-01.
Manuscript, submitted, or to appear
Tverberg Theorems over Discrete Sets of Points.
(Jesús A. De Loera, Thomas Hogan, Frédéric Meunier, Nabil H. Mustafa). Submitted, 2019.
An Application of the Universality Theorem for Tverberg Partitions.
(Imre Barany, Nabil H. Mustafa). Submitted, 2019.
Maximizing Covered Area in a Euclidean Plane with Connectivity Constraint.
(Chien-Chung Huang, Mathieu Mari, Claire Mathieu, Nabil H. Mustafa). Submitted, 2019.
Computing Optimal Epsilon-nets is as Easy as Finding an Unhit Set.
(Nabil H. Mustafa). Submitted, 2019.
Optimal Bounds for VC-dimensions of Unions.
(Monika Csikos, Andrey Kupavskii, Nabil H. Mustafa). Submitted, 2019.
The Discrete Yet Ubiquitous Theorems of Caratheodory, Helly, Sperner, Tucker and Tverberg.
(Jesus de Loera, Xavier Goaoc, Frederic Meunier, Nabil H. Mustafa). Bulletin of the American Mathematical Society, 2019. DOI: https://doi.org/10.1090/bull/1653.