We prove the Strengthened Hanna Neumann Conjecture. We give a more direct cohomological interpretation of the conjecture in terms of “typical” covering maps, and use graph Galois theory to “symmetrize” the conjecture. The conjecture is then related to certain kernel of a morphism of sheaves, and is implied provided these kernels are co-acyclic in the covering cohomology theory. This allows us t...

We study H(p, q)-colorings of graphs, for H a fixed simple graph and p, q natural numbers, a generalization of various other vertex partitioning concepts such as H-covering. An H-cover of a graph G is a local isomorphism between G and H , and the complexity of deciding if an input graph G has an H-cover is still open for many graphs H . In this paper we show that the complexity of H(2p, q)-COLO...

In this paper, we consider the Generalized Bin Covering problem: We are given m bin types, where each bin of type i has profit pi and demand di. Furthermore, there are n items, where item j has size sj . A bin of type i is covered if the set of items assigned to it has total size at least the demand di. In that case, the profit of pi is earned and the objective is to maximize the total profit. ...

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and...

Chang’s Lemma is a widely employed result in additive combinatorics. It gives optimal bounds on the dimension of the large spectrum of probability distributions on nite abelian groups. In this note, we show how Chang’s Lemma and a powerful variant due to Bloom both follow easily from an approximation theorem for probability measures in terms of generalized Riesz products. The latter result invo...

In this thesis we study algorithms for scheduling problems. We investigate semi-online minimum makespan scheduling and generalized bin covering. In online minimum makespan scheduling we are given a set of m machines and n jobs, where each job Jt is specified by a processing time. The jobs arrive one by one and we have to assign them to the machines without any knowledge about future incoming jo...

We introduce the concepts of T-generalized state machines and coverings of products of them. Also some of algebraic properties of them are investigated. Finely some products such as direct sum and sum of Tgeneralized state machines are introduced. An interesting distributive property of cascade product over the sum of Tgeneralized state machines concern to covering of T-generalized state machin...

The classical Erdős-Ko-Rado theorem states that if k ≤ ⌊n/2⌋ then the largest family of pairwise intersecting k-subsets of [n] = {0, 1, . . . , n} is of size ( n−1 k−1 ) . A family of k subsets satisfying this pairwise intersecting property is called an EKR family. We generalize the EKR property and provide asymptotic lower bounds on the size of the largest family A of k-subsets of [n] that sat...

Let G = (V ∪W,E) be an undirected graph, where V ∪W = {1, ..., n} is the vertex set and E = {(i, j) | i, j ∈ V ∪W, i < j} is the edge set. Vertex s = 1 is the source vertex, V is a set of vertices that might be visited, T ⊆ V is a set of vertices that must be visited (s ∈ T ), and W is a set of vertices that must be covered. A symmetric distance matrix C = (cij), defined on E, uses the Euclidea...

The facet of Knapsack ploytope, i.e. convex hull of 0-1 points satisfying a given linear inequality has been presented in this current paper. Such type of facets plays an important role in set covering set partitioning, matroidal-intersection vertex- packing, generalized assignment and other combinatorial problems. Strong covers for facets of Knapsack ploytope has been developed in the first pa...