411 research outputs found
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
A study of large, medium and small scale structures in the topside ionosphere
Alouette and ISIS data were studied for large, medium, and small scale structures in the ionosphere. Correlation was also sought with measurements by other satellites, such as the Atmosphere Explorer C and E and the Dynamic Explorer 2 satellites, of both neutrals and ionization, and with measurements by ground facilities, such as the incoherent scatter radars. Large scale coherent wavelike structures were found from ISIS 2 electron density contours from above the F peak to nearly the satellite altitude. Such structures were also found to correlate with the observation by AE-C below the F peak during a conjunction of the two satellites. Vertical wavefronts found in the upper F region suggest the dominance of diffusion along field lines as well. Also discovered were multiple, evenly spaced field-aligned ducts in the F region that, at low latitudes, extended to the other hemisphere and were in the form of field-aligned sheets in the east-west direction. Low latitude heating events were discovered that could serve as sources for waves in the ionosphere
A morphological study of waves in the thermosphere using DE-2 observations
Theoretical model and data analysis of DE-2 observations for determining the correlation between the neutral wave activity and plasma irregularities have been presented. The relationships between the observed structure of the sources, precipitation and joule heating, and the fluctuations in neutral and plasma parameters are obtained by analyzing two measurements of neutral atmospheric wave activity and plasma irregularities by DE-2 during perigee passes at an altitude on the order of 300 to 350 km over the polar cap. A theoretical model based on thermal nonlinearity (joule heating) to give mode-mode coupling is developed to explore the role of neutral disturbance (winds and gravity waves) on the generation of plasma irregularities
Sampling-based Approximation Algorithms for Multi-stage Stochastic Optimization
Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being #P-complete.
We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed , for a rich class of -stage stochastic linear programs (LPs), we show that, for any probability distribution, for any , one can compute, with high probability, a solution with expected cost at most times the optimal expected cost, in time polynomial in the input size, , and a parameter that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples.
As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage stochastic integer programs, which includes the multi-stage versions of the set cover, vertex cover, multicut on trees, facility location, and multicommodity flow problems
A min-max theorem for the minimum fleet-size problem
A retrospective fleet-sizing problem can be solved via bipartite matching,
where a maximum cardinality matching corresponds to the minimum number of
vehicles needed to cover all trips. We prove a min-max theorem on this minimum
fleet-size problem: the maximum number of pairwise incompatible trips is equal
to the minimum fleet size needed
Hitting Sets when the Shallow Cell Complexity is Small
The hitting set problem is a well-known NP-hard optimization problem in
which, given a set of elements and a collection of subsets, the goal is to find
the smallest selection of elements, such that each subset contains at least one
element in the selection. Many geometric set systems enjoy improved
approximation ratios, which have recently been shown to be tight with respect
to the shallow cell complexity of the set system. The algorithms that exploit
the cell complexity, however, tend to be involved and computationally
intensive. This paper shows that a slightly improved asymptotic approximation
ratio for the hitting set problem can be attained using a much simpler
algorithm: solve the linear programming relaxation, take one initial random
sample from the set of elements with probabilities proportional to the
LP-solution, and, while there is an unhit set, take an additional sample from
it proportional to the LP-solution. Our algorithm is a simple generalization of
the elegant net-finder algorithm by Nabil Mustafa. To analyze this algorithm
for the hitting set problem, we generalize the classic Packing Lemma, and the
more recent Shallow Packing Lemma, to the setting of weighted epsilon-nets.Comment: Accepted by WAOA202
Stochastic Vehicle Routing with Recourse
We study the classic Vehicle Routing Problem in the setting of stochastic
optimization with recourse. StochVRP is a two-stage optimization problem, where
demand is satisfied using two routes: fixed and recourse. The fixed route is
computed using only a demand distribution. Then after observing the demand
instantiations, a recourse route is computed -- but costs here become more
expensive by a factor lambda.
We present an O(log^2 n log(n lambda))-approximation algorithm for this
stochastic routing problem, under arbitrary distributions. The main idea in
this result is relating StochVRP to a special case of submodular orienteering,
called knapsack rank-function orienteering. We also give a better approximation
ratio for knapsack rank-function orienteering than what follows from prior
work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of
approximation for StochVRP, even on star-like metrics on which our algorithm
achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of
Theorem 1.
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