Properties and Algorithms of the KCube Graphs

The KCube interconnection topology was rst introduced in 2010. The KCube graph is a compound graph of a Kautz digraph and hypercubes. Compared with the at- tractive Kautz digraph and well known hypercube graph, the KCube graph could accommodate as many nodes as possible for a given indegree (and outdegree) and the diameter of interconnection networks. However, there are few algorithms designed for the KCube graph. In this thesis, we will concentrate on nding graph theoretical properties of the KCube graph and designing parallel algorithms that run on this network. We will explore several topological properties, such as bipartiteness, Hamiltonianicity, and symmetry property. These properties for the KCube graph are very useful to develop efficient algorithms on this network. We will then study the KCube network from the algorithmic point of view, and will give an improved routing algorithm. In addition, we will present two optimal broadcasting algorithms. They are fundamental algorithms to many applications. A literature review of the state of the art network designs in relation to the KCube network as well as some open problems in this field will also be given

Feedback Numbers of Möbius Ladders

A subset F ⊂ V(G) is called a feedback vertex set if the subgraph G−F is acyclic. The minimum cardinality of a feedback vertex set is called the feedback number of G, which is proposed by Beineke and Vandell [1]. In this paper, we consider a particular topology graph called Möbius ladders M2n. We use f(M2n) to denote the feedback number of M2n. This paper proves that f (M2n) = [n+1/2], n≥3

Structural Controllability of Multi-Agent Systems Subject to Partial Failure

Formation control of multi-agent systems has emerged as a topic of major interest during the last decade, and has been studied from various perspectives using different approaches. This work considers the structural controllability of multi-agent systems with leader-follower architecture. To this end, graphical conditions are first obtained based on the information flow graph of the system. Then, the notions of p-link, q-agent, and joint-(p,q) controllability are introduced as quantitative measures for the controllability of the system subject to failure in communication links or/and agents. Necessary and sufficient conditions for the system to remain structurally controllable in the case of the failure of some of the communication links or/and loss of some agents are derived in terms of the topology of the information flow graph. Moreover, a polynomial-time algorithm for determining the maximum number of failed communication links under which the system remains structurally controllable is presented. The proposed algorithm is analogously extended to the case of loss of agents, using the node-duplication technique. The above results are subsequently extended to the multiple-leader case, i.e., when more than one agent can act as the leader. Then, leader localization problem is investigated, where it is desired to achieve p-link or q-agent controllability in a multi-agent system. This problem is concerned with finding a minimal set of agents whose selection as leaders results in a p-link or q-agent controllable system. Polynomial-time algorithms to find such minimal sets for both undirected and directed information flow graphs are presented

Structural conrollability of multi-agent systems subject to partial failure

Formation control of multi-agent systems has emerged as a topic of major interest during the last decade, and has been studied from various perspectives using different approaches. This work considers the structural controllability of multi-agent systems with leader-follower architecture. To this end, graphical conditions are first obtained based on the information flow graph of the system. Then, the notions of p -link, q- agent, and joint-(p, q ) controllability are introduced as quantitative measures for the controllability of the system subject to failure in communication links or/and agents. Necessary and sufficient conditions for the system to remain structurally controllable in the case of the failure of some of the communication links or/and loss of some agents are derived in terms of the topology of the information flow graph. Moreover, a polynomial-time algorithm for determining the maximum number of failed communication links under which the system remains structurally controllable is presented. The proposed algorithm is analogously extended to the case of loss of agents, using the node-duplication technique. The above results are subsequently extended to the multiple-leader case, i.e., when more than one agent can act as the leader. Then, leader localization problem is investigated, where it is desired to achieve p -link or q -agent controllability in a multi-agent system. This problem is concerned with finding a minimal set of agents whose selection as leaders results in a p -link or q -agent controllable system. Polynomial-time algorithms to find such minimal sets for both undirected and directed information flow graphs are presente

Failure Analysis in Multi-Agent Networks: A Graph-Theoretic Approach

A multi-agent network system consists of a group of dynamic control agents which interact according to a given information flow structure. Such cooperative dynamics over a network may be strongly affected by the removal of network nodes and communication links, thus potentially compromising the functionality of the overall system. The chief purpose of this thesis is to explore and address the challenges of multi-agent cooperative control under various fault and failure scenarios by analyzing the network graph-topology. In the first part, the agents are assumed to evolve according to the linear agreement protocol. Link failures in the network are characterized based on the ability to distinguish the agent dynamics before and after failures. Sufficient topological conditions are provided, under which dynamics of a given agent is distinguishable for distinct digraphs. The second part of this thesis is concerned with the preservation of structural controllability for a multi-agent network under simultaneous link and agent failures. To this end, the previously studied concepts of link and agent controllability degrees are first exploited to provide quantitative measures for the contribution of a particular link or agent to the controllability of the overall network. Next, the case when both communication links and agents in the network can fail simultaneously is considered, and graphical conditions for preservation of controllability are investigated

Feedback Numbers of Goldberg Snark, Twisted Goldberg Snarks and Related Graphs

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. The minimum cardinality of a feedback vertex set is called the feedback number. In this paper, we determine the exact values of the feedback numbers of the Goldberg snarks Gn and its related graphs Gn*, Twisted Goldberg Snarks TGn and its related graphs TGn*. Let f(n) denote the feedback numbers of these graphs, we prove that f(n)=2n+1, for n≥3

Parameterized Complexity Classification for Interval Constraints

Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most $k$ constraints, where $k$ is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen's interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set $A$ of 13 basic comparison relations such as ``precedes'' or ``during'' for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP$(\Gamma)$ for all $\Gamma \subseteq A$. IA is sometimes extended with unions of the relations in $A$ or first-order definable relations over $A$, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP$(A)$ in general, we then consider (parameterized) approximation algorithms and present a factor-$2$ fpt-approximation algorithm

Formally Verified Compositional Algorithms for Factored Transition Systems

Artificial Intelligence (AI) planning and model checking are two disciplines that found wide practical applications. It is often the case that a problem in those two fields concerns a transition system whose behaviour can be encoded in a digraph that models the system's state space. However, due to the very large size of state spaces of realistic systems, they are compactly represented as propositionally factored transition systems. These representations have the advantage of being exponentially smaller than the state space of the represented system. Many problems in AI~planning and model checking involve questions about state spaces, which correspond to graph theoretic questions on digraphs modelling the state spaces. However, existing techniques to answer those graph theoretic questions effectively require, in the worst case, constructing the digraph that models the state space, by expanding the propositionally factored representation of the syste\ m. This is not practical, if not impossible, in many cases because of the state space size compared to the factored representation. One common approach that is used to avoid constructing the state space is the compositional approach, where only smaller abstractions of the system at hand are processed and the given problem (e.g. reachability) is solved for them. Then, a solution for the problem on the concrete system is derived from the solutions of the problem on the abstract systems. The motivation of this approach is that, in the worst case, one need only construct the state spaces of the abstractions which can be exponentially smaller than the state space of the concrete system. We study the application of the compositional approach to two fundamental problems on transition systems: upper-bounding the topological properties (e.g. the largest distance between any two states, i.e. the diameter) of the state spa\ ce, and computing reachability between states. We provide new compositional algorithms to solve both problems by exploiting different structures of the given system. In addition to the use of an existing abstraction (usually referred to as projection) based on removing state space variables, we develop two new abstractions for use within our compositional algorithms. One of the new abstractions is also based on state variables, while the other is based on assignments to state variables. We theoretically and experimentally show that our new compositional algorithms improve the state-of-the-art in solving both problems, upper-bounding state space topological parameters and reachability. We designed the algorithms as well as formally verified them with the aid of an interactive theorem prover. This is the first application that we are aware of, for such a theorem prover based methodology to the design of new algorithms in either AI~planning or model checking