767 research outputs found
Multi-Goal Feasible Path Planning Using Ant Colony Optimization
A new algorithm for solving multi-goal planning problems in the presence of obstacles is introduced. We extend ant colony optimization (ACO) from its well-known application, the traveling salesman problem (TSP), to that of multi-goal feasible path planning for inspection and surveillance applications. Specifically, the ant colony framework is combined with a sampling-based point-to-point planning algorithm; this is compared with two successful sampling-based multi-goal planning algorithms in an obstacle-filled two-dimensional environment. Total mission time, a function of computational cost and the duration of the planned mission, is used as a basis for comparison. In our application of interest, autonomous underwater inspections, the ACO algorithm is found to be the best-equipped for planning in minimum mission time, offering an interior point in the tradeoff between computational complexity and optimality.United States. Office of Naval Research (Grant N00014-06-10043
The random link approximation for the Euclidean traveling salesman problem
The traveling salesman problem (TSP) consists of finding the length of the
shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where
the cities are distributed randomly and independently in a d-dimensional unit
hypercube. Working with periodic boundary conditions and inspired by a
remarkable universality in the kth nearest neighbor distribution, we find for
the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with
beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive
analytical predictions for these quantities using the random link
approximation, where the lengths between cities are taken as independent random
variables. From the ``cavity'' equations developed by Krauth, Mezard and
Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3,
numerical results show that the random link approximation is a good one, with a
discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we
argue that the approximation is exact up to O(1/d^2) and give a conjecture for
beta_E(d), in terms of a power series in 1/d, specifying both leading and
subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte
The Dynamic Multi-objective Multi-vehicle Covering Tour Problem
This work introduces a new routing problem called the Dynamic Multi-Objective Multi-vehicle Covering Tour Problem (DMOMCTP). The DMOMCTPs is a combinatorial optimization problem that represents the problem of routing multiple vehicles to survey an area in which unpredictable target nodes may appear during execution. The formulation includes multiple objectives that include minimizing the cost of the combined tour cost, minimizing the longest tour cost, minimizing the distance to nodes to be covered and maximizing the distance to hazardous nodes. This study adapts several existing algorithms to the problem with several operator and solution encoding variations. The efficacy of this set of solvers is measured against six problem instances created from existing Traveling Salesman Problem instances which represent several real countries. The results indicate that repair operators, variable length solution encodings and variable-length operators obtain a better approximation of the true Pareto front
Two new Probability inequalities and Concentration Results
Concentration results and probabilistic analysis for combinatorial problems
like the TSP, MWST, graph coloring have received much attention, but generally,
for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example).
Here, we prove two probability inequalities which generalize and strengthen
Martingale inequalities. The inequalities provide the tools to deal with more
general heavy-tailed and inhomogeneous distributions for combinatorial
problems. We prove a wide range of applications - in addition to the TSP, MWST,
graph coloring, we also prove more general results than known previously for
concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random
projection theorem. It is hoped that the strength of the inequalities will
serve many more purposes.Comment: 3
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