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We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than…
Ridesharing services play an essential role in modern transportation, which significantly reduces traffic congestion and exhaust pollution. In the ridesharing problem, improving the sharing rate between riders can not only save the travel…
We discuss several extension properties of matroids and polymatroids and their application as necessary conditions for the existence of different matroid representations, namely linear, folded linear, algebraic, and entropic…
We provide optimal solutions to an institution that has distributional objectives when choosing from a set of applications based on merit (or priority). For example, in college admissions, administrators may want to admit a diverse class in…
The basis exchange axiom has been a driving force in the development of matroid theory. However, the axiom gives only a local characterization of the relation of bases, which is a major stumbling block to further progress, and providing a…
We prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of $t$-path ideals of a complete multipartite graph.
Current techniques for generating a knowledge space, such as QUERY, guarantees that the resulting structure is closed under union, but not that it satisfies wellgradedness, which is one of the defining conditions for a learning space. We…
Recommendation, information retrieval, and other information access systems pose unique challenges for investigating and applying the fairness and non-discrimination concepts that have been developed for studying other machine learning…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for…
Multiparty computation is raising importance because it's primary objective is to replace any trusted third party in the distributed computation. This work presents two multiparty shuffling protocols where each party, possesses a private…
Finding community structures in networks is important in network science, technology, and applications. To date, most algorithms that aim to find community structures only focus either on unipartite or bipartite networks. A unipartite…
Imitation is a basic updating mechanism for strategy evolution in structured populations, determining how individuals sample social information and translate it into behavioral changes. Higher-order networks, such as hypergraphs, generalize…
In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of…
Conjunctive Hierarchical Secret Sharing (CHSS) is a type of secret sharing that divides participants into multiple distinct hierarchical levels, with each level having a specific threshold. An authorized subset must simultaneously meet the…
We consider a problem, which we call secure grouping, of dividing a number of parties into some subsets (groups) in the following manner: Each party has to know the other members of his/her group, while he/she may not know anything about…
The family of Information Dispersal Algorithms is applied to distributed systems for secure and reliable storage and transmission. In comparison with perfect secret sharing it achieves a significantly smaller memory overhead and better…
Besides the advantages derived from the ever present communication properties, it increases the attack surface of a network as well. As industrial protocols and systems were not designed with security in mind, spectacular attacks on…
A preferential arrangement of a finite set is an ordered partition. Associated with each such ordered partition is a chain of subsets or blocks endowed with a linear order. The chain may be split into sections by the introduction of a…
In 1968, Liu described the problem of securing documents in a shared secret project. In an example, at least six out of eleven participating scientists need to be present to open the lock securing the secret documents. Shamir proposed a…