32 research outputs found
Vehicle Routing using the Sweep Algorithm In Parallel
This paper presents a parallel version of the Sweep Algorithm, a heuristic solution to the smgle-termma] vehicle routing problem The Sweep Algorithm uses a duster-first route-second approach in finding a near-optimal set of routes. The clusters of delivery points are formed by using the terminal as the center and moving around it m a sweeping fashion After a duster is defined, the route is found with a traveling salesperson algorithm.
The parallel version begins the sweep at different angles and performs both forward and backward sweeps. Each node handles a sweep and returns information concerning total distance traveled to the host. The host then derides which node has the best routes and requests the specific information.
The traveling salesperson algorithm is the nearest neighbor method. This starts the route at the point in the duster nearest the terminal and proceeds by visiting the nearest unvisited point. Since this procedure is called many times in the course of finding all the routes, the quick but good results that this method yields were attractive.
Also discussed are other parallel implementations of this problem and ideas for further research
Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints
Multidimensional optimization problems where the objective function and the
constraints are multiextremal non-differentiable Lipschitz functions (with
unknown Lipschitz constants) and the feasible region is a finite collection of
robust nonconvex subregions are considered. Both the objective function and the
constraints may be partially defined. To solve such problems an algorithm is
proposed, that uses Peano space-filling curves and the index scheme to reduce
the original problem to a H\"{o}lder one-dimensional one. Local tuning on the
behaviour of the objective function and constraints is used during the work of
the global optimization procedure in order to accelerate the search. The method
neither uses penalty coefficients nor additional variables. Convergence
conditions are established. Numerical experiments confirm the good performance
of the technique.Comment: 29 pages, 5 figure
Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP
Given a metric space on n points, an {\alpha}-approximate universal algorithm
for the Steiner tree problem outputs a distribution over rooted spanning trees
such that for any subset X of vertices containing the root, the expected cost
of the induced subtree is within an {\alpha} factor of the optimal Steiner tree
cost for X. An {\alpha}-approximate differentially private algorithm for the
Steiner tree problem takes as input a subset X of vertices, and outputs a tree
distribution that induces a solution within an {\alpha} factor of the optimal
as before, and satisfies the additional property that for any set X' that
differs in a single vertex from X, the tree distributions for X and X' are
"close" to each other. Universal and differentially private algorithms for TSP
are defined similarly. An {\alpha}-approximate universal algorithm for the
Steiner tree problem or TSP is also an {\alpha}-approximate differentially
private algorithm. It is known that both problems admit O(logn)-approximate
universal algorithms, and hence O(log n)-approximate differentially private
algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation
ratio achievable for the universal Steiner tree problem and the universal TSP,
matching the known upper bounds. Our lower bound for the Steiner tree problem
holds even when the algorithm is allowed to output a more general solution of a
distribution on paths to the root.Comment: 14 page
Star Routing: Between Vehicle Routing and Vertex Cover
We consider an optimization problem posed by an actual newspaper company,
which consists of computing a minimum length route for a delivery truck, such
that the driver only stops at street crossings, each time delivering copies to
all customers adjacent to the crossing. This can be modeled as an abstract
problem that takes an unweighted simple graph and a subset of
edges and asks for a shortest cycle, not necessarily simple, such that
every edge of has an endpoint in the cycle.
We show that the decision version of the problem is strongly NP-complete,
even if is a grid graph. Regarding approximate solutions, we show that the
general case of the problem is APX-hard, and thus no PTAS is possible unless P
NP. Despite the hardness of approximation, we show that given any
-approximation algorithm for metric TSP, we can build a
-approximation algorithm for our optimization problem, yielding a
concrete -approximation algorithm.
The grid case is of particular importance, because it models a city map or
some part of it. A usual scenario is having some neighborhood full of
customers, which translates as an instance of the abstract problem where almost
every edge of is in . We model this property as , and
for these instances we give a -approximation algorithm,
for any , provided that the grid is sufficiently big.Comment: Accepted to the 12th Annual International Conference on Combinatorial
Optimization and Applications (COCOA'18
Two-photon-induced stretchable graphene supercapacitors
Direct laser writing with an ultrashort laser beam pulses has emerged as a cost-effective single step technology for realizing high spatial resolution features of three-dimensional structures in confined footprints with potential for large area fabrication. Here we present the two-photon direct laser writing technology to develop high-performance stretchable biomimetic three-dimensional micro-supercapacitors with the fractal electrode distance down to 1 ”m. With multilayered graphene oxide films, we show the charge transfer capability enhanced by order of 102 while the energy storage density exceeds the results in current lithium-ion batteries. The stretchability and the volumetric capacitance are increased to 150% and 86 mF/cm3 (0.181 mF/cm2), respectively. This additive nanofabrication method is highly desirable for the development of self-sustainable stretchable energy storage integrated with wearable technologies. The flexible and stretchable energy storage with a high energy density opens the new opportunity for on-chip sensing, imaging, and monitoring
Sixteen space-filling curves and traversals for d-dimensional cubes and simplices
This article describes sixteen different ways to traverse d-dimensional space
recursively in a way that is well-defined for any number of dimensions. Each of
these traversals has distinct properties that may be beneficial for certain
applications. Some of the traversals are novel, some have been known in
principle but had not been described adequately for any number of dimensions,
some of the traversals have been known. This article is the first to present
them all in a consistent notation system. Furthermore, with this article, tools
are provided to enumerate points in a regular grid in the order in which they
are visited by each traversal. In particular, we cover: five discontinuous
traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton
indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and
Inside-out traversal; two discontinuous traversals based on subdividing
simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected
traversal; five continuous traversals based on subdividing cubes into 2^d
subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa
Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four
continuous traversals based on subdividing cubes into 3^d subcubes: the Peano
curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these
traversals are self-similar in the sense that the traversal in each of the
subcubes or subsimplices of a cube or simplex, on any level of recursive
subdivision, can be obtained by scaling, translating, rotating, reflecting
and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line
Stochastic and dynamic vehicle routing with general demand and interarrival time distributions
Includes bibliographical references (p. 38-40).Supported by the National Science Foundation. DDM-9014751 Supported by Draper Laboratories and the UPS Foundation.Dimitris J. Bertsimas, Garrett van Ryzin
Reordering Rows for Better Compression: Beyond the Lexicographic Order
Sorting database tables before compressing them improves the compression
rate. Can we do better than the lexicographical order? For minimizing the
number of runs in a run-length encoding compression scheme, the best approaches
to row-ordering are derived from traveling salesman heuristics, although there
is a significant trade-off between running time and compression. A new
heuristic, Multiple Lists, which is a variant on Nearest Neighbor that trades
off compression for a major running-time speedup, is a good option for very
large tables. However, for some compression schemes, it is more important to
generate long runs rather than few runs. For this case, another novel
heuristic, Vortex, is promising. We find that we can improve run-length
encoding up to a factor of 3 whereas we can improve prefix coding by up to 80%:
these gains are on top of the gains due to lexicographically sorting the table.
We prove that the new row reordering is optimal (within 10%) at minimizing the
runs of identical values within columns, in a few cases.Comment: to appear in ACM TOD
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Set covering with our eyes closed
Given a universe of elements and a weighted collection of subsets of , the universal set cover problem is to a priori map each element to a set containing such that any set is covered by S(X)=\cup_{u\in XS(u). The aim is to find a mapping such that the cost of is as close as possible to the optimal set cover cost for . (Such problems are also called oblivious or a priori optimization problems.) Unfortunately, for every universal mapping, the cost of can be times larger than optimal if the set is adversarially chosen. In this paper we study the performance on average, when is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (nonmetric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multicut and disc-covering problems and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors