27 research outputs found
Shortest Reconfiguration of Sliding Tokens on a Caterpillar
Suppose that we are given two independent sets I_b and I_r of a graph such
that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|.
Then, the sliding token problem is to determine whether there exists a sequence
of independent sets which transforms I_b into I_r so that each independent set
in the sequence results from the previous one by sliding exactly one token
along an edge in the graph. The sliding token problem is one of the
reconfiguration problems that attract the attention from the viewpoint of
theoretical computer science. The reconfiguration problems tend to be
PSPACE-complete in general, and some polynomial time algorithms are shown in
restricted cases. Recently, the problems that aim at finding a shortest
reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy
for the complexity of finding the shortest sequence has been shown, that is, it
is in P, NP-complete, or PSPACE-complete in certain conditions. In general,
even if it is polynomial time solvable to decide whether two instances are
reconfigured with each other, it can be NP-complete to find a shortest sequence
between them. Namely, finding a shortest sequence between two independent sets
can be more difficult than the decision problem of reconfigurability between
them. In this paper, we show that the problem for finding a shortest sequence
between two independent sets is polynomial time solvable for some graph classes
which are subclasses of the class of interval graphs. More precisely, we can
find a shortest sequence between two independent sets on a graph G in
polynomial time if either G is a proper interval graph, a trivially perfect
graph, or a caterpillar. As far as the authors know, this is the first
polynomial time algorithm for the shortest sliding token problem for a graph
class that requires detours
Sliding Tokens on a Cactus
Given two independent sets I and J of a graph G, imagine that a token (coin) is placed on each vertex in I. Then, the Sliding Token problem asks if one could transforms I to J using a sequence of elementary steps, where each step requires sliding a token from one vertex to one of its neighbors, such that the resulting set of vertices where tokens are placed still remains independent. In this paper, we describe a polynomial-time algorithm for solving Sliding Token in case the graph G is a cactus. Our algorithm is designed based on two observations. First, all structures that forbid the existence of a sequence of token slidings between I and J, if exist, can be found in polynomial time. A no-instance may be easily deduced using this characterization. Second, without such forbidden structures, a sequence of token slidings between I and J does exist
Shortest Reconfiguration of Matchings
Imagine that unlabelled tokens are placed on the edges of a graph, such that
no two tokens are placed on incident edges. A token can jump to another edge if
the edges having tokens remain independent. We study the problem of determining
the distance between two token configurations (resp., the corresponding
matchings), which is given by the length of a shortest transformation. We give
a polynomial-time algorithm for the case that at least one of the two
configurations is not inclusion-wise maximal and show that otherwise, the
problem admits no polynomial-time sublogarithmic-factor approximation unless P
= NP. Furthermore, we show that the distance of two configurations in bipartite
graphs is fixed-parameter tractable parameterized by the size of the
symmetric difference of the source and target configurations, and obtain a
-factor approximation algorithm for every if
additionally the configurations correspond to maximum matchings. Our two main
technical tools are the Edmonds-Gallai decomposition and a close relation to
the Directed Steiner Tree problem. Using the former, we also characterize those
graphs whose corresponding configuration graphs are connected. Finally, we show
that deciding if the distance between two configurations is equal to a given
number is complete for the class , and deciding if the diameter of
the graph of configurations is equal to is -hard.Comment: 31 pages, 3 figure
Heterogeneous Self-Reconfiguring Robotics
Self-reconfiguring (SR) robots are modular systems that can autonomously change shape, or reconfigure, for increased versatility and adaptability in unknown environments. In this thesis, we investigate planning and control for systems of non-identical modules, known as heterogeneous SR robots. Although previous approaches rely on module homogeneity as a critical property, we show that the planning complexity of fundamental algorithmic problems in the heterogeneous case is equivalent to that of systems with identical modules. Primarily, we study the problem of how to plan shape changes while considering the placement of specific modules within the structure. We characterize this key challenge in terms of the amount of free space available to the robot and develop a series of decentralized reconfiguration planning algorithms that assume progressively more severe free space constraints and support reconfiguration among obstacles. In addition, we compose our basic planning techniques in different ways to address problems in the related task domains of positioning modules according to function, locomotion among obstacles, self-repair, and recognizing the achievement of distributed goal-states. We also describe the design of a novel simulation environment, implementation results using this simulator, and experimental results in hardware using a planar SR system called the Crystal Robot. These results encourage development of heterogeneous systems. Our algorithms enhance the versatility and adaptability of SR robots by enabling them to use functionally specialized components to match capability, in addition to shape, to the task at hand
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Reconfiguration of Subgraphs Satisfying Specific Properties
Tohoku University周暁課