799 research outputs found
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
A hierarchical approach to improve the ant colony optimization algorith
The ant colony optimization algorithm (ACO) is a fast heuristic-based method for finding favorable solutions to the traveling salesman problem (TSP). When the data set reaches larger values however, the ACO runtime increases dramatically. As a result, clustering nodes into groups is an effective way to reduce the size of the problem while leveraging the advantages of the ACO algorithm. The method for recombining groups of nodes is explored by treating the graph as a hierarchy of clusters, and modifying the original ACO heuristic to operate on a hypergraph. This method of using hierarchical clustering is significantly faster than the original ACO algorithm, even when normal clustering techniques are applied, while producing improved tour lengths
An interacting replica approach applied to the traveling salesman problem
We present a physics inspired heuristic method for solving combinatorial
optimization problems. Our approach is specifically motivated by the desire to
avoid trapping in metastable local minima- a common occurrence in hard problems
with multiple extrema. Our method involves (i) coupling otherwise independent
simulations of a system ("replicas") via geometrical distances as well as (ii)
probabilistic inference applied to the solutions found by individual replicas.
The {\it ensemble} of replicas evolves as to maximize the inter-replica
correlation while simultaneously minimize the local intra-replica cost function
(e.g., the total path length in the Traveling Salesman Problem within each
replica). We demonstrate how our method improves the performance of rudimentary
local optimization schemes long applied to the NP hard Traveling Salesman
Problem. In particular, we apply our method to the well-known "-opt"
algorithm and examine two particular cases- and . With the aid of
geometrical coupling alone, we are able to determine for the optimum tour
length on systems up to cities (an order of magnitude larger than the
largest systems typically solved by the bare opt). The probabilistic
replica-based inference approach improves even further and determines
the optimal solution of a problem with cities and find tours whose total
length is close to that of the optimal solutions for other systems with a
larger number of cities.Comment: To appear in SAI 2016 conference proceedings 12 pages,17 figure
Solving the Traveling Salesman Problem with release dates via branch and cut
In this paper we study the Traveling Salesman Problem with release dates (TSP-rd) and completion time
minimization. The TSP-rd considers a single vehicle and a set of customers that must be served exactly
once with goods that arrive to the depot over time, during the planning horizon. The time at which each
requested good arrives is called release date and it is known in advance. The vehicle can perform multiple
routes, however, it cannot depart to serve a customer before the associated release date. Thus, the release
date of the customers in each route must not be greater than the starting time of the route. The objective
is to determine a set of routes for the vehicle, starting and ending at the depot, where the completion time
needed to serve all customers is minimized. We propose a new Integer Linear Programming model and develop
a branch and cut algorithm with tailored enhancements to improve its performance. The algorithm proved
to be able to significantly reduce the computation times when compared to a compact formulation tackled
using a commercial mathematical programming solver, obtaining 24 new optimal solutions on benchmark
instances with up to 30 customers within one hour. We further extend the benchmark to instances with up to
50 customers where the algorithm proved to be efficient. Building upon these results, the proposed model is
adapted to new TSP-rd variants (Capacitated and Prize-Collecting TSP), with different objectives: completion
time minimization and traveling distance minimization. To the best of our knowledge, our work is the first
in-depth study to report extensive results for the TSP-rd through a branch and cut, establishing a baseline and
providing insights for future approaches. Overall, the approach proved to be very effective and gives a flexible
framework for several variants, opening the discussion about formulations, algorithms and new benchmark
instances
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