649 research outputs found
Reconfiguring k-Path Vertex Covers
A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The K-PATH VERTEX COVER RECONFIGURATION (K-PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of K-PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: TS, TJ, and TAR. The problem for k=2, known as the VERTEX COVER RECONFIGURATION (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes can be extended for K-PVCR. In particular, we prove a complexity dichotomy for K-PVCR on general graphs: on those whose maximum degree is three (and even planar), the problem is PSPACE-complete, while on those whose maximum degree is two (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for K-PVCR on trees under each of TJ and TAR. Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest
Reconfiguring k-path vertex covers
A vertex subset of a graph is called a -path vertex cover if every
path on vertices in contains at least one vertex from . The
\textsc{-Path Vertex Cover Reconfiguration (-PVCR)} problem asks if one
can transform one -path vertex cover into another via a sequence of -path
vertex covers where each intermediate member is obtained from its predecessor
by applying a given reconfiguration rule exactly once. We investigate the
computational complexity of \textsc{-PVCR} from the viewpoint of graph
classes under the well-known reconfiguration rules: ,
, and . The problem for , known as the
\textsc{Vertex Cover Reconfiguration (VCR)} problem, has been well-studied in
the literature. We show that certain known hardness results for \textsc{VCR} on
different graph classes including planar graphs, bounded bandwidth graphs,
chordal graphs, and bipartite graphs, can be extended for \textsc{-PVCR}. In
particular, we prove a complexity dichotomy for \textsc{-PVCR} on general
graphs: on those whose maximum degree is (and even planar), the problem is
-complete, while on those whose maximum degree is (i.e.,
paths and cycles), the problem can be solved in polynomial time. Additionally,
we also design polynomial-time algorithms for \textsc{-PVCR} on trees under
each of and . Moreover, on paths, cycles, and
trees, we describe how one can construct a reconfiguration sequence between two
given -path vertex covers in a yes-instance. In particular, on paths, our
constructed reconfiguration sequence is shortest.Comment: 29 pages, 4 figures, to appear in WALCOM 202
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
A Game-theoretic Formulation of the Homogeneous Self-Reconfiguration Problem
In this paper we formulate the homogeneous two- and three-dimensional
self-reconfiguration problem over discrete grids as a constrained potential
game. We develop a game-theoretic learning algorithm based on the
Metropolis-Hastings algorithm that solves the self-reconfiguration problem in a
globally optimal fashion. Both a centralized and a fully distributed algorithm
are presented and we show that the only stochastically stable state is the
potential function maximizer, i.e. the desired target configuration. These
algorithms compute transition probabilities in such a way that even though each
agent acts in a self-interested way, the overall collective goal of
self-reconfiguration is achieved. Simulation results confirm the feasibility of
our approach and show convergence to desired target configurations.Comment: 8 pages, 5 figures, 2 algorithm
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
Distributed Vertex Cover Reconfiguration
Reconfiguration schedules, i.e., sequences that gradually transform one solution of a problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration schedule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers. We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfiguration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover.
To set a baseline for batch reconfiguration, we show that for graphs belonging to one of the classes {{cycles, trees, forests, chordal, cactus, even-hole-free, claw-free}}, there are schedules that use O(?^{-1}) batches and incur only a 1+? multiplicative increase in cost over the best sequential schedules. Our main contribution is to compute such batch schedules in a distributed setting O(?^{-1} {log^*} n) rounds, which we also show to be tight. Further, we show that once we step out of these graph classes we face a very different situation. There are graph classes on which no efficient distributed algorithm can obtain the best (or almost best) existing schedule. Moreover, there are classes of bounded degree graphs which do not admit any reconfiguration schedules without incurring a large multiplicative increase in the cost at all
TS-Reconfiguration of -Path Vertex Covers in Caterpillars for
A -path vertex cover (-PVC) of a graph is a vertex subset such that each path on vertices in contains at least one member of . Imagine that a token is placed on each vertex of a -PVC. Given two -PVCs of a graph , the -Path Vertex Cover Reconfiguration (-PVCR) under Token Sliding () problem asks if there is a sequence of -PVCs between and where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be -complete even for planar graphs of maximum degree and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, for , we present a polynomial-time algorithm that solves -PVCR under for caterpillars (i.e., trees formed by attaching leaves to a path)
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