Edge-Orders

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time

Hardness and Algorithms for Rainbow Connectivity

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon$ > 0, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connectivity, where the bound depends only on $\epsilon$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also pre sented

Experience-Based Planning with Sparse Roadmap Spanners

We present an experienced-based planning framework called Thunder that learns to reduce computation time required to solve high-dimensional planning problems in varying environments. The approach is especially suited for large configuration spaces that include many invariant constraints, such as those found with whole body humanoid motion planning. Experiences are generated using probabilistic sampling and stored in a sparse roadmap spanner (SPARS), which provides asymptotically near-optimal coverage of the configuration space, making storing, retrieving, and repairing past experiences very efficient with respect to memory and time. The Thunder framework improves upon past experience-based planners by storing experiences in a graph rather than in individual paths, eliminating redundant information, providing more opportunities for path reuse, and providing a theoretical limit to the size of the experience graph. These properties also lead to improved handling of dynamically changing environments, reasoning about optimal paths, and reducing query resolution time. The approach is demonstrated on a 30 degrees of freedom humanoid robot and compared with the Lightning framework, an experience-based planner that uses individual paths to store past experiences. In environments with variable obstacles and stability constraints, experiments show that Thunder is on average an order of magnitude faster than Lightning and planning from scratch. Thunder also uses 98.8% less memory to store its experiences after 10,000 trials when compared to Lightning. Our framework is implemented and freely available in the Open Motion Planning Library.Comment: Submitted to ICRA 201

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure