Percolation on nonunimodular transitive graphs

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with $p_c=1$ on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.Comment: Published at http://dx.doi.org/10.1214/009117906000000494 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the intersection of tolerance and cocomparability graphs.

It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. The conjecture has been proved under some – rather strong – structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. In this article we prove that the above conjecture is true for every graph G, whose tolerance representation satisfies a slight assumption; note here that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. This assumption on the representation is guaranteed by a wide variety of graph classes; for example, our results immediately imply the correctness of the conjecture for complements of triangle-free graphs (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are algorithmic, in the sense that, given a tolerance representation R of a graph G, we describe an algorithm to transform R into a bounded tolerance representation R  ∗  of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic and Monma, it suffices to prove our conjecture. In addition, there already exists evidence in the literature that our conjecture is true

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

In this paper, we present a hierarchical path planning framework called SG-RL (subgoal graphs-reinforcement learning), to plan rational paths for agents maneuvering in continuous and uncertain environments. By "rational", we mean (1) efficient path planning to eliminate first-move lags; (2) collision-free and smooth for agents with kinematic constraints satisfied. SG-RL works in a two-level manner. At the first level, SG-RL uses a geometric path-planning method, i.e., Simple Subgoal Graphs (SSG), to efficiently find optimal abstract paths, also called subgoal sequences. At the second level, SG-RL uses an RL method, i.e., Least-Squares Policy Iteration (LSPI), to learn near-optimal motion-planning policies which can generate kinematically feasible and collision-free trajectories between adjacent subgoals. The first advantage of the proposed method is that SSG can solve the limitations of sparse reward and local minima trap for RL agents; thus, LSPI can be used to generate paths in complex environments. The second advantage is that, when the environment changes slightly (i.e., unexpected obstacles appearing), SG-RL does not need to reconstruct subgoal graphs and replan subgoal sequences using SSG, since LSPI can deal with uncertainties by exploiting its generalization ability to handle changes in environments. Simulation experiments in representative scenarios demonstrate that, compared with existing methods, SG-RL can work well on large-scale maps with relatively low action-switching frequencies and shorter path lengths, and SG-RL can deal with small changes in environments. We further demonstrate that the design of reward functions and the types of training environments are important factors for learning feasible policies.Comment: 20 page

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

On the Evaluation of RDF Distribution Algorithms Implemented over Apache Spark

Querying very large RDF data sets in an efficient manner requires a sophisticated distribution strategy. Several innovative solutions have recently been proposed for optimizing data distribution with predefined query workloads. This paper presents an in-depth analysis and experimental comparison of five representative and complementary distribution approaches. For achieving fair experimental results, we are using Apache Spark as a common parallel computing framework by rewriting the concerned algorithms using the Spark API. Spark provides guarantees in terms of fault tolerance, high availability and scalability which are essential in such systems. Our different implementations aim to highlight the fundamental implementation-independent characteristics of each approach in terms of data preparation, load balancing, data replication and to some extent to query answering cost and performance. The presented measures are obtained by testing each system on one synthetic and one real-world data set over query workloads with differing characteristics and different partitioning constraints.Comment: 16 pages, 3 figure

On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed