23 research outputs found
Independence and matching numbers of some token graphs
Let be a graph of order and let . The -token
graph of , is the graph whose vertices are the -subsets of
, where two vertices are adjacent in whenever their symmetric
difference is an edge of . We study the independence and matching numbers of
. We present a tight lower bound for the matching number of
for the case in which has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite -token
graphs, and determine the exact value for some graphs.Comment: 16 pages, 4 figures. Third version is a major revision. Some proofs
were corrected or simplified. New references adde
Finding a needle in an exponential haystack: Discrete RRT for exploration of implicit roadmaps in multi-robot motion planning
We present a sampling-based framework for multi-robot motion planning which
combines an implicit representation of a roadmap with a novel approach for
pathfinding in geometrically embedded graphs tailored for our setting. Our
pathfinding algorithm, discrete-RRT (dRRT), is an adaptation of the celebrated
RRT algorithm for the discrete case of a graph, and it enables a rapid
exploration of the high-dimensional configuration space by carefully walking
through an implicit representation of a tensor product of roadmaps for the
individual robots. We demonstrate our approach experimentally on scenarios of
up to 60 degrees of freedom where our algorithm is faster by a factor of at
least ten when compared to existing algorithms that we are aware of.Comment: Kiril Solovey and Oren Salzman contributed equally to this pape
Motion Planning for Unlabeled Discs with Optimality Guarantees
We study the problem of path planning for unlabeled (indistinguishable)
unit-disc robots in a planar environment cluttered with polygonal obstacles. We
introduce an algorithm which minimizes the total path length, i.e., the sum of
lengths of the individual paths. Our algorithm is guaranteed to find a solution
if one exists, or report that none exists otherwise. It runs in time
, where is the number of robots and is the total
complexity of the workspace. Moreover, the total length of the returned
solution is at most , where OPT is the optimal solution cost. To
the best of our knowledge this is the first algorithm for the problem that has
such guarantees. The algorithm has been implemented in an exact manner and we
present experimental results that attest to its efficiency
Independence and matching number for some token graphs
Let G be a graph of order n and let k ∈ {1, . . . , n−1}. The k-token graph
Fk(G) of G is the graph whose vertices are the k-subsets of V (G), where
two vertices are adjacent in Fk(G) whenever their symmetric difference
is an edge of G. We study the independence and matching numbers of
Fk(G). We present a tight lower bound for the matching number of Fk(G)
for the case in which G has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite ktoken
graphs, and determine the exact value for some graphs