Related papers: Supermodularity in Unweighted Graph Optimization I…
This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses…
We consider the problem of maximizing submodular functions; while this problem is known to be NP-hard, several numerically efficient local search techniques with approximation guarantees are available. In this paper, we propose a novel…
In this note we extend the spectral theorem for bimodules to the higher rank graph C*-algebra context. Under the assumption that the graph is row finite and has no sources, we show that a bimodule over a natural abelian subalgebra is…
This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate…
We prove that every graph of rankwidth at least $72r$ contains an induced subgraph whose minimum balanced cutrank is at least $r$, which implies a vertex subset where every balanced separation has $\mathbb{F}_2$-cutrank at least $r$. This…
Graph-based semi-supervised learning has proven to be an effective approach for query-focused multi-document summarization. The problem of previous semi-supervised learning is that sentences are ranked without considering the higher level…
Knowledge graphs have emerged to be promising datastore candidates for context augmentation during Retrieval Augmented Generation (RAG). As a result, techniques in graph representation learning have been simultaneously explored alongside…
We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups. We compare characterizations of superreflexivity in terms of diamond graphs and binary trees. We show that there exist…
We study alternating minimization for matrix completion in the simplest possible setting: completing a rank-one matrix from a revealed subset of the entries. We bound the asymptotic convergence rate by the variational characterization of…
We present a method for learning max-weight matching predictors in bipartite graphs. The method consists of performing maximum a posteriori estimation in exponential families with sufficient statistics that encode permutations and data…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
Submodular functions have been studied extensively in machine learning and data mining. In particular, the optimization of submodular functions over the integer lattice (integer submodular functions) has recently attracted much interest,…
The problem of combinatorially determining the rank of the 3-dimensional bar-joint {\em rigidity matroid} of a graph is an important open problem in combinatorial rigidity theory. Maxwell's condition states that the edges of a graph $G=(V,…
This article provides a comprehensive exploration of submodular maximization problems, focusing on those subject to uniform and partition matroids. Crucial for a wide array of applications in fields ranging from computer science to systems…
Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…
A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound…
Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This…
Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as high-dimensional complex integrals. Motivated by the need for a systematic study of the asymptotic…
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…