12 research outputs found
The Complexity of Rerouting Shortest Paths
The Shortest Path Reconfiguration problem has as input a graph G (with unit
edge lengths) with vertices s and t, and two shortest st-paths P and Q. The
question is whether there exists a sequence of shortest st-paths that starts
with P and ends with Q, such that subsequent paths differ in only one vertex.
This is called a rerouting sequence.
This problem is shown to be PSPACE-complete. For claw-free graphs and chordal
graphs, it is shown that the problem can be solved in polynomial time, and that
shortest rerouting sequences have linear length. For these classes, it is also
shown that deciding whether a rerouting sequence exists between all pairs of
shortest st-paths can be done in polynomial time. Finally, a polynomial time
algorithm for counting the number of isolated paths is given.Comment: The results on claw-free graphs, chordal graphs and isolated paths
have been added in version 2 (april 2012). Version 1 (September 2010) only
contained the PSPACE-hardness result. (Version 2 has been submitted.
Independent Set Reconfiguration in Cographs
We study the following independent set reconfiguration problem, called
TAR-Reachability: given two independent sets and of a graph , both
of size at least , is it possible to transform into by adding and
removing vertices one-by-one, while maintaining an independent set of size at
least throughout? This problem is known to be PSPACE-hard in general. For
the case that is a cograph (i.e. -free graph) on vertices, we show
that it can be solved in time , and that the length of a shortest
reconfiguration sequence from to is bounded by , if such a
sequence exists.
More generally, we show that if is a graph class for which (i)
TAR-Reachability can be solved efficiently, (ii) maximum independent sets can
be computed efficiently, and which satisfies a certain additional property,
then the problem can be solved efficiently for any graph that can be obtained
from a collection of graphs in using disjoint union and complete join
operations. Chordal graphs are given as an example of such a class
Rerouting shortest paths in planar graphs
A rerouting sequence is a sequence of shortest st-paths such that consecutive
paths differ in one vertex. We study the the Shortest Path Rerouting Problem,
which asks, given two shortest st-paths P and Q in a graph G, whether a
rerouting sequence exists from P to Q. This problem is PSPACE-hard in general,
but we show that it can be solved in polynomial time if G is planar. To this
end, we introduce a dynamic programming method for reconfiguration problems.Comment: submitte
Subset sum problems with digraph constraints
We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees
When Nash Meets Stackelberg
Motivated by international energy trade between countries with
profit-maximizing domestic producers, we analyze Nash games played among
leaders of Stackelberg games (\NASP). We prove it is both -hard to
decide if the game has a pure-strategy (\PNE) or a mixed-strategy Nash
equilibrium (\MNE). We then provide a finite algorithm that computes exact
\MNEs for \NASPs when there is at least one, or returns a certificate if no
\MNE exists. To enhance computational speed, we introduce an inner
approximation hierarchy that increasingly grows the description of each
Stackelberg leader feasible region. Furthermore, we extend the algorithmic
framework to specifically retrieve a \PNE if one exists. Finally, we provide
computational tests on a range of \NASPs instances inspired by international
energy trades.Comment: 40 Pages and a computational appendix. Code is available on gitHu
Motion planning with pulley, rope, and baskets
We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported into the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given threshold. We prove that it is Pi-2-p-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions