2,644 research outputs found
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
A review of the Tabu Search Literature on Traveling Salesman Problems
The Traveling Salesman Problem (TSP) is one of the most widely studied problems inrncombinatorial optimization. It has long been known to be NP-hard and hence research onrndeveloping algorithms for the TSP has focused on approximate methods in addition to exactrnmethods. Tabu search is one of the most widely applied metaheuristic for solving the TSP. Inrnthis paper, we review the tabu search literature on the TSP, point out trends in it, and bringrnout some interesting research gaps in this literature.
Uncapacitated Flow-based Extended Formulations
An extended formulation of a polytope is a linear description of this
polytope using extra variables besides the variables in which the polytope is
defined. The interest of extended formulations is due to the fact that many
interesting polytopes have extended formulations with a lot fewer inequalities
than any linear description in the original space. This motivates the
development of methods for, on the one hand, constructing extended formulations
and, on the other hand, proving lower bounds on the sizes of extended
formulations.
Network flows are a central paradigm in discrete optimization, and are widely
used to design extended formulations. We prove exponential lower bounds on the
sizes of uncapacitated flow-based extended formulations of several polytopes,
such as the (bipartite and non-bipartite) perfect matching polytope and TSP
polytope. We also give new examples of flow-based extended formulations, e.g.,
for 0/1-polytopes defined from regular languages. Finally, we state a few open
problems
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Combinatorial optimization and metaheuristics
Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
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