Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex. We derive a convex representation of this intersection to calculate the ranges of the coordinates. Secondly, we propose two new methods for obtaining r...

A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomial-time approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTAS...

We give an O(n) algorithm for finding the PQ tree of a consecutive-ones matrix if a consecutive-ones ordering is given, where n is the number of columns and an O(n)-space representation of the matrix is given. We use this to obtain the modular decomposition of permutation graphs and two-dimensional partial orders inO(n) time when their compact representation with two linear orders is given. Mor...

Submodular functions are common in combinatorics; examples include the cut capacity function of a graph and the rank function of a matroid. The submodular function minimization problem generalizes the classical minimum cut problem and also contains a number of other combinatorial optimization problems as special cases. In this thesis, we study submodular function minimization and two related pr...

In recent years, the theory of group representations has greatly benefited from a new approach provided by the topology of singular spaces, namely intersection cohomology (IC ) theory. Let G be a connected reductive algebraic group over an algebraically closed field (which will be assumed to be C unless otherwise specified) ; although G itself is clearly a non-singular variety, the study of the...

We consider a 2-dimensional representation of the Hecke algebra $H(G_7, u)$, where $G_7$ is the complex reflection group and $u$ is the set of indeterminates $u = (x_1,x_2,y_1,y_2,y_3,z_1,z_2,z_3)$. After specializing the indetrminates to non zero complex numbers, we then determine a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of the repre...

A (n, l, γ)-sharing set family of sizem is a family of sets S1, . . . , Sm ⊆ [n] s.t. each set has size l and each pair of sets shares at most γ elements. We let m (n, l, γ) denote the maximum size of any such set family and we consider the following question: How large can m (n, l, γ) be? (n, l, γ)-sharing set families have a rich set of applications including the construction of pseudorandom ...

For a family F of sets, let ex(F) := {A : A is an extremal intersecting sub-family of F}. The Erd®s-Ko-Rado (EKR) Theorem states that {A ∈ ( [n] r ) : 1 ∈ A} ∈ ex( ( [n] r ) ) if r ≤ n/2. The Hilton-Milner (HM) Theorem states that if r ≤ n/2 and A is a nontrivial intersecting sub-family of ( [n] r ) then |A| ≤ |{A ∈ ( [n] r ) : 1 ∈ A,A ∩ [2, r + 1] 6= ∅} ∪ {[2, r + 1]}|; hence {{A ∈ ( [n] r ) :...

Let 2 ≤ d ≤ k be fixed and n be sufficiently large. Suppose that G is a collection of k-element subsets of an n-element set, and |G| > ( n−1 k−1 ) . Then G contains d sets with union of size at most 2k and empty intersection. This verifies a conjecture of the first author for large n.

A private set intersection protocol consists of two parties, a Sender and a Receiver, each with a secret input set. The protocol aims to have the Receiver output an intersection of the two sets while keeping the elements in the sets secret. This thesis thoroughly analyzes four recently published set intersection protocols, where it explains each protocol and checks whether it satisfies its corr...