The Full Degree Spanning Tree Problem

Given a graph G, we study the problem of finding a spanning tree T that maximizes the number of vertices of full degree; that is, the number of vertices whose degree in T equals its degree in G. We prove a few general bounds and then analyze this parameter on various classes of graphs including grid graphs, hypercubes, and random regular graphs. We also explore a related problem that focuses on maximizing the number of leaves in a spanning tree of a graph

Recognizing Partial Cubes in Quadratic Time

We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n^2), improving previous O(nm)-time solutions.Comment: 25 pages, five figures. This version significantly expands previous versions, including a new report on an implementation of the algorithm and experiments with i

Some Optimally Adaptive Parallel Graph Algorithms on EREW PRAM Model

The study of graph algorithms is an important area of research in computer science, since graphs offer useful tools to model many real-world situations. The commercial availability of parallel computers have led to the development of efficient parallel graph algorithms. Using an exclusive-read and exclusive-write (EREW) parallel random access machine (PRAM) as the computation model with a fixed number of processors, we design and analyze parallel algorithms for seven undirected graph problems, such as, connected components, spanning forest, fundamental cycle set, bridges, bipartiteness, assignment problems, and approximate vertex coloring. For all but the last two problems, the input data structure is an unordered list of edges, and divide-and-conquer is the paradigm for designing algorithms. One of the algorithms to solve the assignment problem makes use of an appropriate variant of dynamic programming strategy. An elegant data structure, called the adjacency list matrix, used in a vertex-coloring algorithm avoids the sequential nature of linked adjacency lists. Each of the proposed algorithms achieves optimal speedup, choosing an optimal granularity (thus exploiting maximum parallelism) which depends on the density or the number of vertices of the given graph. The processor-(time)2 product has been identified as a useful parameter to measure the cost-effectiveness of a parallel algorithm. We derive a lower bound on this measure for each of our algorithms

Aspects of practical implementations of PRAM algorithms

The PRAM is a shared memory model of parallel computation which abstracts away from inessential engineering details. It provides a very simple architecture independent model and provides a good programming environment. Theoreticians of the computer science community have proved that it is possible to emulate the theoretical PRAM model using current technology. Solutions have been found for effectively interconnecting processing elements, for routing data on these networks and for distributing the data among memory modules without hotspots. This thesis reviews this emulation and the possibilities it provides for large scale general purpose parallel computation. The emulation employs a bridging model which acts as an interface between the actual hardware and the PRAM model. We review the evidence that such a scheme crn achieve scalable parallel performance and portable parallel software and that PRAM algorithms can be optimally implemented on such practical models. In the course of this review we presented the following new results: 1. Concerning parallel approximation algorithms, we describe an NC algorithm for finding an approximation to a minimum weight perfect matching in a complete weighted graph. The algorithm is conceptually very simple and it is also the first NC-approximation algorithm for the task with a sub-linear performance ratio. 2. Concerning graph embedding, we describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the 2-dimcnsional mesh. In the embeddings the maximum distance from a leaf to the root of the tree is asymptotically optimally short. The embeddings facilitate efficient implementation of many PRAM algorithms on networks employing these graphs as interconnection networks. 3. Concerning bulk synchronous algorithmics, we describe scalable transportable algorithms for the following three commonly required types of computation; balanced tree computations. Fast Fourier Transforms and matrix multiplications

Algorithms for Game-Theoretic Environments

Game Theory constitutes an appropriate way for approaching the Internet and modelling situations where participants interact with each other, such as networking, online auctions and search engine’s page ranking. Mechanism Design deals with the design of private-information games and attempts implementing desired social choices in a strategic setting. This thesis studies how the efficiency of a system degrades due to the selfish behaviour of its agents, expressed in terms of the Price of Anarchy (PoA). Our objective is to design mechanisms with improved PoA, or to determine the exact value of the PoA for existing mechanisms for two well-known problems, Auctions and Network Cost-Sharing Design. We study three different settings of auctions, combinatorial auction, multi- unit auction and bandwidth allocation. The combinatorial auction constitutes a fundamental resource allocation problem that involves the interaction of selfish agents in competition for indivisible goods. Although it is well-known that by using the VCG mechanism the selfishness of the agents does not affect the efficiency of the system, i.e. the social welfare is maximised, this mechanism cannot generally be applied in computationally tractable time. In practice, several simple auctions (lacking some nice properties of the VCG) are used, such as the generalised second price auction on AdWords, the simultaneous ascending price auction for spectrum allocation, and the independent second-price auction on eBay. The latter auction is of particular interest in this thesis. Precisely, we give tight bounds on the PoA when the goods are sold in independent and simultaneous first-price auctions, where the highest bidder gets the item and pays her own bid. Then, we generalise our results to a class of auctions that we call bid-dependent auctions, where the goods are also sold in independent and simultaneous auctions and further the payment of each bidder is a function of her bid, even if she doesn’t get the item. Overall, we show that the first-price auction is optimal among all bid-dependent auctions. The multi-unit auction is a special case of combinatorial auction where all items are identical. There are many variations: the discriminatory auction, the uniform price auction and the Vickrey multi-unit auction. In all those auctions, the goods are allocated to the highest marginal bids, and their difference lies on the pricing scheme. Our focus is on the discriminatory auction, which can be seen as the variant of the first-price auction adjusted to multi-unit auctions. The bandwidth allocation is equivalent to auctioning divisible resources. Allocating network resources, like bandwidth, among agents is a canonical problem in the network optimisation literature. A traditional model for this problem was proposed by Kelly [1997], where each agent receives a fraction of the resource proportional to her bid and pays her own bid. We complement the PoA bounds known in the literature and give tight bounds for a more general case. We further show that this mechanism is optimal among a wider class of mechanisms. We further study design issues for network games: given a rooted undirected graph with nonnegative edge costs, a set of players with terminal vertices need to establish connectivity with the root. Each player selects a path and the global objective is to minimise the cost of the used edges. The cost of an edge may represent infrastructure cost for establishing connectivity or renting expense, and needs to be covered by the users. There are several ways to split the edge cost among its users and this is dictated by a cost-sharing protocol. Naturally, it is in the players best interest to choose paths that charge them with small cost. The seminal work of Chen et al. [2010] was the first to address design questions for this game. They thoroughly studied the PoA for the following informational assumptions. i) The designer has full knowledge of the instance, that is, she knows both the network topology and the players’ terminals. ii) The designer has no knowledge of the underlying graph. Arguably, there are situations where the former assumption is too optimistic while the latter is too pessimistic. We propose a model that lies in the middle-ground; the designer has prior knowledge of the underlying metric, but knows nothing about the positions of the terminals. Her goal is to process the graph and choose a universal cost-sharing protocol that has low PoA against all possible requested subsets. The main question is to what extent prior knowledge of the underlying metric can help in the design. We first demonstrate that there exist graph metrics where knowledge of the underlying metric can dramatically improve the performance of good network cost-sharing design. However, in our main technical result, we show that there exist graph metrics for which knowing the underlying metric does not help and any universal protocol matches the bound of Chen et al. [2010] which ignores the graph metric. We further study the stochastic and Bayesian games where the players choose their terminals according to a probability distribution. We showed that in the stochastic setting there exists a priority protocol that achieves constant PoA, whereas the PoA under the the Bayesian setting can be very high for any cost- sharing protocol satisfying some natural properties

Polychromatic colorings of certain subgraphs of complete graphs and maximum densities of substructures of a hypercube

If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. For H, K, subsets of the vertex set V(Qd) of the d-cube Qd, K is an exact copy of H if there is an automorphism of Qd sending H to K. For a positive integer, d, and a configuration in Qd, H, we define λ(H,d) as the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, λ(H,d) for several configurations in Q2, Q3, and Q4, and an infinite family of configurations. Strong connections exist with extensions Ramsey numbers for cycles in a graph, counting sequences with certain properties, inducibility of graphs, and we determine the inducibility of two vertex disjoint edges in the family of bipartite graphs

Problems in extremal and combinatorial geometry

This thesis deals with three families of optimization problems: (1) Euclidean optimization problems on random point sets; (2) independent sets in hypergraphs; and (3) packings in point lattices. First, we consider bounds on several monochromatic and bichromatic optimization problems including minimum matching, minimum spanning trees, and the travelling salesman problem. Many of these problems lend themselves to representations in terms of hierarchically separated trees | trees with uniform branching factor and depth, and having edge weights exponential in the depth of the edge in the tree. In the second part, we consider the independent set problem on uniform hypergraphs, in anticipation of applying it to the third part, packing problems on point lattices. In these problems we wish to select a subset of points from an n n ::: n grid avoiding particular patterns. We also study several generalizations of these problems that have not been handled previously.M.S., Computer Science -- Drexel University, 201