36 research outputs found
Covering Pairs in Directed Acyclic Graphs
The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a
classical problem that provides a clear and simple mathematical formulation for
several applications in different areas and that has an efficient algorithmic
solution. In this paper, we study the computational complexity of two
constrained variants of Minimum Path Cover motivated by the recent introduction
of next-generation sequencing technologies in bioinformatics. The first problem
(MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum
cardinality set of paths "covering" all the vertices such that both vertices of
each pair belong to the same path. For this problem, we show that, while it is
NP-hard to compute if there exists a solution consisting of at most three
paths, it is possible to decide in polynomial time whether a solution
consisting of at most two paths exists. The second problem (MaxRPSP), given a
DAG and a set of pairs of vertices, asks for a path containing the maximum
number of the given pairs of vertices. We show its NP-hardness and also its
W[1]-hardness when parametrized by the number of covered pairs. On the positive
side, we give a fixed-parameter algorithm when the parameter is the maximum
overlapping degree, a natural parameter in the bioinformatics applications of
the problem
Patrolling a path connecting a set of points with unbalanced frequencies of visits
Patrolling consists of scheduling perpetual movements of a collection of mobile robots, so that each point of the environment is regularly revisited by any robot in the collection. In previous research, it was assumed that all points of the environment needed to be revisited with the same minimal frequency. In this paper we study efficient patrolling protocols for points located on a path, where each point may have a different constraint on frequency of visits. The problem of visiting such divergent points was recently posed by GÄ
sieniec et al. in [14], where the authors study protocols using a single robot patrolling a set of n points located in nodes of a complete graph and in Euclidean spaces. The focus in this paper is on patrolling with two robots. We adopt a scenario in which all points to be patrolled are located on a line. We provide several approximation algorithms concluding with the best currently known 3 -approximation
Patrolling on Dynamic Ring Networks
We study the problem of patrolling the nodes of a network collaboratively by
a team of mobile agents, such that each node of the network is visited by at
least one agent once in every time units, with the objective of
minimizing the idle time . While patrolling has been studied previously
for static networks, we investigate the problem on dynamic networks with a
fixed set of nodes, but dynamic edges. In particular, we consider
1-interval-connected ring networks and provide various patrolling algorithms
for such networks, for or agents. We also show almost matching
lower bounds that hold even for the best starting configurations. Thus, our
algorithms achieve close to optimal idle time. Further, we show a clear
separation in terms of idle time, for agents that have prior knowledge of the
dynamic networks compared to agents that do not have such knowledge. This paper
provides the first known results for collaborative patrolling on dynamic
graphs
Using Minimum Path Cover to Boost Dynamic Programming on DAGs : Co-linear Chaining Extended
Peer reviewe
Path Planning in O/1/infinity Weighted Regions with Applications
Path Planning in O/1/infinity Weighted Regions with Application