556 research outputs found
Self-Assembly of DNA Graphs and Postman Tours
DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and the Windy Postman Problem
Parameterized Directed -Chinese Postman Problem and Arc-Disjoint Cycles Problem on Euler Digraphs
In the Directed -Chinese Postman Problem (-DCPP), we are given a
connected weighted digraph and asked to find non-empty closed directed
walks covering all arcs of such that the total weight of the walks is
minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128)
asked for the parameterized complexity of -DCPP when is the parameter.
We prove that the -DCPP is fixed-parameter tractable.
We also consider a related problem of finding arc-disjoint directed
cycles in an Euler digraph, parameterized by . Slivkins (ESA 2003) showed
that this problem is W[1]-hard for general digraphs. Generalizing another
result by Slivkins, we prove that the problem is fixed-parameter tractable for
Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler
digraphs remains W[1]-hard even for Euler digraphs
Efficient Multi-Robot Coverage of a Known Environment
This paper addresses the complete area coverage problem of a known
environment by multiple-robots. Complete area coverage is the problem of moving
an end-effector over all available space while avoiding existing obstacles. In
such tasks, using multiple robots can increase the efficiency of the area
coverage in terms of minimizing the operational time and increase the
robustness in the face of robot attrition. Unfortunately, the problem of
finding an optimal solution for such an area coverage problem with multiple
robots is known to be NP-complete. In this paper we present two approximation
heuristics for solving the multi-robot coverage problem. The first solution
presented is a direct extension of an efficient single robot area coverage
algorithm, based on an exact cellular decomposition. The second algorithm is a
greedy approach that divides the area into equal regions and applies an
efficient single-robot coverage algorithm to each region. We present
experimental results for two algorithms. Results indicate that our approaches
provide good coverage distribution between robots and minimize the workload per
robot, meanwhile ensuring complete coverage of the area.Comment: In proceedings of IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS), 201
Constant-factor approximations for Capacitated Arc Routing without triangle inequality
Given an undirected graph with edge costs and edge demands, the Capacitated
Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity
vehicles so as to satisfy all demands. Constant-factor polynomial-time
approximation algorithms were proposed for CARP with triangle inequality, while
CARP was claimed to be NP-hard to approximate within any constant factor in
general. Correcting this claim, we show that any factor {\alpha} approximation
for CARP with triangle inequality yields a factor {\alpha} approximation for
the general CARP
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
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