Motioning connected subgraphs into a graph

In this paper we study connected subgraphs and how to motion them inside a connected graph preserving the connectivity. We determine completely the group of movements.Comment: 17 pages, 18 figure

An algorithm with improved complexity for pebble motion/multi-agent path finding on trees

The pebble motion on trees (PMT) problem consists in finding a feasible sequence of moves that repositions a set of pebbles to assigned target vertices. This problem has been widely studied because, in many cases, the more general Multi-Agent path finding (MAPF) problem on graphs can be reduced to PMT. We propose a simple and easy to implement procedure, which finds solutions of length O(knc + n^2), where n is the number of nodes, $k$ is the number of pebbles, and c the maximum length of corridors in the tree. This complexity result is more detailed than the current best known result O(n^3), which is equal to our result in the worst case, but does not capture the dependency on c and k

Feasibility of Motion Planning on Acyclic and Strongly Connected Directed Graphs

Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature for it allows to concentrate on the combinatorial problem abstracting from geometric considerations. In this paper, we consider motion planning over directed graphs, which are of interest for asymmetric communication networks. Directed graphs generalize undirected graphs, while introducing a new source of complexity to the motion planning problem: moves are not reversible. We first consider the class of acyclic directed graphs and show that the feasibility can be solved in time linear in the product of the number of vertices and the number of arcs. We then turn to strongly connected directed graphs. We first prove a structural theorem for decomposing strongly connected directed graphs into strongly biconnected components.Based on the structural decomposition, we give an algorithm for the feasibility of motion planning on strongly connected directed graphs, and show that it can also be decided in time linear in the product of the number of vertices and the number of arcs.Comment: 19 pages, 9 figures, algorithm2e.st

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

Optimal Pebble motion on a Tree

AbstractIn this paper we consider the following pebble coordination problem. Consider a tree with n vertices and k pebbles located at distinct vertices of the tree. Each pebble can be moved from its current position to an adjacent unoccupied vertex. Among the k pebbles, one distinguished pebble has been assigned a destination. We give an O(n5) algorithm for the problem of designing the shortest sequence of moves that takes the distinguished pebble from its original position to its destination. Our algorithm improves the running time of the best previously presented algorithm that needed to solve O(n6) min-cost flow problems on graphs of size O(n). Our algorithm does not resort to reduction to flow but is instead based on a novel dynamic programming approach

Optimal Pebble Motion on a Tree

In this paper we consider the following pebble coordination problem. Consider a tree with n vertices and k pebbles located at distinct vertices of the tree. Each pebble can be moved from its current position to an adjacent unoccupied vertex. Among the k pebbles, one distinguished pebbles has been assigned a destination. We give an O(n 5 ) algorithm for the problem of designing the shortest sequence of moves that takes the distinguished pebble from its original position to its destination. Our algorithm improves the running time of the best previously presented algorithm that needed to solve O(n 6 ) min-cost ow problems on graphs of size O(n). Our algorithm does not resort to reduction to ow but is instead based on a novel dynamic programming approach. A preliminary version of this paper appeared as \A New Approach to Optimal Motion Planning on Trees with Obstacles" in Proceedings of 4{European Symposium on Algorithms (LNCS 1136), pp. 529-545. This work was partially supporte..