16,493 research outputs found
Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics
Simple heuristics often show a remarkable performance in practice for
optimization problems. Worst-case analysis often falls short of explaining this
performance. Because of this, "beyond worst-case analysis" of algorithms has
recently gained a lot of attention, including probabilistic analysis of
algorithms.
The instances of many optimization problems are essentially a discrete metric
space. Probabilistic analysis for such metric optimization problems has
nevertheless mostly been conducted on instances drawn from Euclidean space,
which provides a structure that is usually heavily exploited in the analysis.
However, most instances from practice are not Euclidean. Little work has been
done on metric instances drawn from other, more realistic, distributions. Some
initial results have been obtained by Bringmann et al. (Algorithmica, 2013),
who have used random shortest path metrics on complete graphs to analyze
heuristics.
The goal of this paper is to generalize these findings to non-complete
graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path
metric is constructed by drawing independent random edge weights for each edge
in the graph and setting the distance between every pair of vertices to the
length of a shortest path between them with respect to the drawn weights. For
such instances, we prove that the greedy heuristic for the minimum distance
maximum matching problem, the nearest neighbor and insertion heuristics for the
traveling salesman problem, and a trivial heuristic for the -median problem
all achieve a constant expected approximation ratio. Additionally, we show a
polynomial upper bound for the expected number of iterations of the 2-opt
heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201
Tackling Group-Tree Matching in Large Scale Group Communications
As a mechanism to support group communications, multicasting faces a serious state scalability problem when there are large numbers of groups in the network: lots of resources (e.g., memory to maintain group state information) and control overhead (e.g., multicast tree setup and maintenance) are required to manage the groups. Recently, an efficient solution called aggregated multicast is proposed [8]. In this approach, groups are assigned to proper trees and multiple groups can share one delivery tree. A key problem in aggregated multicast is group-tree matching (i.e., matching groups to trees). In this paper, we investigate this group-tree matching problem. We first formally define the problem, and formulate two versions of the problem: static and dynamic. We analyze the static version of the problem and prove that it is NP-complete. To tackle this hard problem, we propose three algorithms: one optimal (using Linear Integer Programming, or ILP), one near-optimal (using Greedy method), and one pseudo-dynamic algorithm. For the dynamic version, we present a general heuristic on-line grouptree matching algorithm. Simulation studies are conducted to compare the three algorithms for the static version. Our results show that Greedy algorithm is a feasible solution to the static problem and its performance is very close the ILP optimal solution, while pseudo-dynamic algorithm is a good heuristic for many cases where Greedy does not work well. We also evaluate the performance of the heuristic online algorithm, and show that it is a practical solution to the dynamic on-line group-tree matching problem
Some recent results in the analysis of greedy algorithms for assignment problems
We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems
Fast Hierarchical Clustering and Other Applications of Dynamic Closest Pairs
We develop data structures for dynamic closest pair problems with arbitrary
distance functions, that do not necessarily come from any geometric structure
on the objects. Based on a technique previously used by the author for
Euclidean closest pairs, we show how to insert and delete objects from an
n-object set, maintaining the closest pair, in O(n log^2 n) time per update and
O(n) space. With quadratic space, we can instead use a quadtree-like structure
to achieve an optimal time bound, O(n) per update. We apply these data
structures to hierarchical clustering, greedy matching, and TSP heuristics, and
discuss other potential applications in machine learning, Groebner bases, and
local improvement algorithms for partition and placement problems. Experiments
show our new methods to be faster in practice than previously used heuristics.Comment: 20 pages, 9 figures. A preliminary version of this paper appeared at
the 9th ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1998, pp.
619-628. For source code and experimental results, see
http://www.ics.uci.edu/~eppstein/projects/pairs
Highly Scalable Algorithms for Robust String Barcoding
String barcoding is a recently introduced technique for genomic-based
identification of microorganisms. In this paper we describe the engineering of
highly scalable algorithms for robust string barcoding. Our methods enable
distinguisher selection based on whole genomic sequences of hundreds of
microorganisms of up to bacterial size on a well-equipped workstation, and can
be easily parallelized to further extend the applicability range to thousands
of bacterial size genomes. Experimental results on both randomly generated and
NCBI genomic data show that whole-genome based selection results in a number of
distinguishers nearly matching the information theoretic lower bounds for the
problem
Maximizing System Throughput Using Cooperative Sensing in Multi-Channel Cognitive Radio Networks
In Cognitive Radio Networks (CRNs), unlicensed users are allowed to access
the licensed spectrum when it is not currently being used by primary users
(PUs). In this paper, we study the throughput maximization problem for a
multi-channel CRN where each SU can only sense a limited number of channels. We
show that this problem is strongly NP-hard, and propose an approximation
algorithm with a factor at least where is a system
parameter reflecting the sensing capability of SUs across channels and their
sensing budgets. This performance guarantee is achieved by exploiting a nice
structural property of the objective function and constructing a particular
matching. Our numerical results demonstrate the advantage of our algorithm
compared with both a random and a greedy sensing assignment algorithm
Shunting passenger trains: getting ready for departure
In this paper we consider the problem of shunting train units on a railway station. Train units arrive at and depart from the station according to a given train schedule and in between the units may have to be stored at the station. The assignment of arriving to departing train units (called matching) and the scheduling of the movements to realize this matching is called shunting. The goal is to realize the shunting using a minimal number of shunt movements.\ud
For a restricted version of this problem an ILP approach has been presented in the literature. In this paper, we consider the general shunting problem and derive a greedy heuristic approach and an exact solution method based on dynamic programming. Both methods are flexible in the sense that they allow the incorporation of practical planning rules and may be extended to cover additional requirements from practice
The edge-disjoint path problem on random graphs by message-passing
We present a message-passing algorithm to solve the edge disjoint path
problem (EDP) on graphs incorporating under a unique framework both traffic
optimization and path length minimization. The min-sum equations for this
problem present an exponential computational cost in the number of paths. To
overcome this obstacle we propose an efficient implementation by mapping the
equations onto a weighted combinatorial matching problem over an auxiliary
graph. We perform extensive numerical simulations on random graphs of various
types to test the performance both in terms of path length minimization and
maximization of the number of accommodated paths. In addition, we test the
performance on benchmark instances on various graphs by comparison with
state-of-the-art algorithms and results found in the literature. Our
message-passing algorithm always outperforms the others in terms of the number
of accommodated paths when considering non trivial instances (otherwise it
gives the same trivial results). Remarkably, the largest improvement in
performance with respect to the other methods employed is found in the case of
benchmarks with meshes, where the validity hypothesis behind message-passing is
expected to worsen. In these cases, even though the exact message-passing
equations do not converge, by introducing a reinforcement parameter to force
convergence towards a sub optimal solution, we were able to always outperform
the other algorithms with a peak of 27% performance improvement in terms of
accommodated paths. On random graphs, we numerically observe two separated
regimes: one in which all paths can be accommodated and one in which this is
not possible. We also investigate the behaviour of both the number of paths to
be accommodated and their minimum total length.Comment: 14 pages, 8 figure
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