A generalization of totally unimodular and network matrices.

In this thesis we discuss possible generalizations of totally unimodular and network matrices. Our purpose is to introduce new classes of matrices that preserve the advantageous properties of these well-known matrices. In particular, our focus is on the polyhedral consequences of totally unimodular matrices, namely we look for matrices that can ensure vertices that are scalable to an integral vector by an integer k. We argue that simply generalizing the determinantal structure of totally unimodular matrices does not suffice to achieve this goal and one has to extend the range of values the inverses of submatrices can contain. To this end, we define k-regular matrices. We show that k-regularity is a proper generalization of total unimodularity in polyhedral terms, as it guarantees the scalability of vertices. Moreover, we prove that the k-regularity of a matrix is necessary and sufficient for substituting mod-k cuts for rank-1 Chvatal-Gomory cuts. In the second part of the thesis we introduce binet matrices, an extension of network matrices to bidirected graphs. We provide an algorithm to calculate the columns of a binet matrix using the underlying graphical structure. Using this method, we prove some results about binet matrices and demonstrate that several interesting classes of matrices are binet. We show that binet matrices are 2-regular, therefore they provide half-integral vertices for a polyhedron with a binet constraint matrix and integral right hand side vector. We also prove that optimization on such a polyhedron can be carried out very efficiently, as there exists an extension of the network simplex method for binet matrices. Furthermore, the integer optimization with binet matrices is equivalent to solving a matching problem. We also describe the connection of k-regular and binet matrices to other parts of combinatorial optimization, notably to matroid theory and regular vectorspaces

Nash welfare, valuated matroids, and gross substitutes

We study computational aspects of equilibria and fair division problems with a focus on demand and valuation functions that satisfy the (weak) gross substitutes property. We study the Arrow-Debreu exchange market model with divisible goods where agents’ demands satisfy the weak gross substitutes (WGS) property. We give an auction algorithm that obtains an approximate market equilibrium for WGS demands. Previously, such algorithms were known only for restricted classes of WGS demands. We also derive the implications of our technique for spending-restricted market equilibrium for budget-separable piecewise linear concave (budget-SPLC) utilities. Spending-restricted equilibrium was introduced as a continuous relaxation of the Nash SocialWelfare (NSW) problem. Next, we present the first polynomial-time constant-factor approximation algorithm for the NSW problem under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems. They include as special cases assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first polynomial-time constant-factor approximation algorithm for the asymmetric NSW problem under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant. We examine the Matroid Based Valuation (MBV) conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015). It asserts that every (discrete) gross substitute valuation is a matroid based valuation—a valuation obtained from weighted matroid rank functions by repeated applications of merge and endowment operations. Each matroid based valuation turns out to be an endowment of some Rado valuation. By introducing complete classes of valuated matroids, we exhibit a family of valuations that are gross substitutes but not endowed Rado valuations. This refutes the MBV conjecture. The family is defined via sparse paving matroids

Sublinear Computation Paradigm

This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as “fast,” but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants