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A Parallel Architecture for the Generalized Traveling Salesman Problem

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Title: A Parallel Architecture for the Generalized Traveling Salesman Problem


1
A Parallel Architecture for the Generalized
Traveling Salesman Problem
  • Max Scharrenbroich
  • AMSC 663 Mid-Year Progress Report
  • Advisor Dr. Bruce L. Golden
  • R. H. Smith School of Business

2
Presentation Overview
  • Background and Review
  • GTSP Review
  • Applications
  • Problem Formulation and Algorithms
  • Review of Project Objectives
  • Review of mrOX GA
  • Parallelism in the mrOX GA
  • Status Summary and Future Work

Review gt
3
Review of the GTSP
  • The Generalized Traveling Salesman Problem
    (GTSP)
  • Variation of the well-known traveling salesman
    problem.
  • A set of nodes is partitioned into a number of
    clusters.
  • Objective Find a minimum-cost tour visiting
    exactly one node in each cluster.
  • Example on the following slides

4
GTSP Example
  • Start with a set of nodes or locations to visit.

5
GTSP Example (continued)
  • Partition the nodes into clusters.

6
GTSP Example (continued)
  • Find the minimum tour visiting each cluster.

Applications gt
7
Applications
  • The GTSP has many real-world applications in the
    field of routing
  • Mailbox collection and stochastic vehicle
    routing.
  • Warehouse order picking with multiple stock
    locations.
  • Airport selection and routing for courier planes.

Formulation and Algorithms gt
8
Problem Formulation and Algorithms
  • The GTSP can be formulated as a 0-1 Integer
    Linear Program (ILP).
  • The GTSP is an NP-hard problem.
  • There are algorithms for solving the GTSP to
    optimality, but these algorithms eventually
    suffer from combinatorial explosion (gt90
    clusters).
  • There are several successful heuristic approaches
    to the GTSP.
  • In this project I will focus on the mrOX Genetic
    Algorithm (J. Silberholz and B.L. Golden, 2007).

Review of Project Objectivesgt
9
Review of Project Objectives
  • Develop a generic software architecture and
    framework for parallelizing serial heuristics for
    combinatorial optimization.
  • Extend the framework to host the serial mrOX GA
    and the GTSP problem class.
  • Investigate the performance of the parallel
    implementation of the mrOX GA on large instances
    of the GTSP (gt 90 clusters).

Overview gt
10
Presentation Overview
  • Background and Review
  • Review of mrOX GA
  • mrOX Crossover
  • Local Search 2-opt and 1-swap
  • Mutation
  • Parallelism in the mrOX GA
  • Status Summary and Future Work

Diagram of mrOX GA gt
11
Review of mrOX GA
Start
Pop 1
Pop 2

Pop 7
Merge
Post-Merge
End
Crossover and Local Search gt
12
Review of mrOX GA
mrOX Crossover gt
13
Review of mrOX Crossover
Example gt
14
Example of mrOX Crossover
Complexity gt
15
Complexity of mrOX Crossover
Local Search gt
16
Local Search 2-opt 1-swap
Mutation gt
17
Mutation
(1) Randomly generate cut points.
Overview gt
18
Presentation Overview
  • Background and Review
  • Review of mrOX GA
  • Parallelism in the mrOX GA
  • Low Level Parallelism
  • Concurrent Exploration
  • Cellular GA Inspired Parallel Cooperation
  • Status Summary and Future Work

Low Level Parallelism gt
19
Low-Level Parallelism in the mrOX GA
  • For rOX and mrOX computational loading can be
    estimated (slide 13) and therefore crossover of
    individuals could be load balanced over a number
    of processors.
  • The local search improvement phase (multiple
    cycles of 2-opt followed by 1-swap) in the
    post-merge phase is not deterministic and would
    be difficult to load balance.
  • While improving execution time, load balancing
    the crossover and local search would introduce
    inefficiencies.
  • Tuning the load balancing would be problem
    dependent and time consuming .

Type 3 Parallelism gt
20
Type 3 Parallelism in the mrOX GA
  • Genetic algorithms are amenable to parallelism
    via concurrent exploration.
  • Cooperation between processes can be implemented
    to ensure diversity while maintaining
    intensification.

Parallel Cooperation w/ Mesh gt
21
Parallel Cooperation with Mesh Topology
  • Inspired by cellular genetic algorithms (cGAs),
    where individuals in a population only interact
    with nearest neighbors.
  • Processes cooperate over a toroidal mesh
    topology.
  • Ensures diversity while maintaining
    intensification.

Each process has four neighbors.
Processes periodically exchange the best
solutions with neighbors.
High-quality solutions diffuse through the
population.
Overview gt
22
Presentation Overview
  • Background and Review
  • Review of mrOX GA
  • Parallelism in the mrOX GA
  • Status Summary and Future Work

Status Summary gt
23
Status Summary
  • Investigated different ways of using parallelism
    in the mrOX GA.
  • Learned about cellular genetic algorithms (cGAs)
    as a motivation for parallel cooperation schemes.
  • Completed a preliminary software design and began
    coding.
  • Coded and ran an intermediate test application to
    validate mesh communication pattern.

Future Work gt
24
Future Work
  • Obtain a user account on the Deepthought cluster.
  • Continue with design and coding.
  • Performance testing of parallel implementation.

End gt
25
References
  • Crainic, T.G. and Toulouse, M. Parallel
    Strategies for Meta-Heuristics. Fleet Management
    and Logistics, 205-251.
  • L. Davis. Applying Adaptive Algorithms to
    Epistatic Domains. Proceeding of the
    International Joint Conference on Artificial
    Intelligence, 162-164, 1985.
  • M. Fischetti, J.J. Salazar-Gonzalez, P. Toth. A
    branch-and-cut algorithm for the symmetric
    generalized traveling salesman problem.
    Operations Research 45 (3) 378394, 1997.
  • G. Laporte, A. Asef-Vaziri, C. Sriskandarajah.
    Some Applications of the Generalized Traveling
    Salesman Problem. Journal of the Operational
    Research Society 47 1461-1467, 1996.
  • C.E. Noon. The generalized traveling salesman
    problem. Ph. D. Dissertation, University of
    Michigan, 1988.
  • C.E. Noon. A Lagrangian based approach for the
    asymmetric generalized traveling salesman
    problem. Operations Research 39 (4) 623-632,
    1990.
  • J.P. Saksena. Mathematical model of scheduling
    clients through welfare agencies. CORS Journal 8
    185-200, 1970.
  • J. Silberholz and B.L. Golden. The Generalized
    Traveling Salesman Problem A New Genetic
    Algorithm Approach. Operations Research/Computer
    Science Interfaces Series 37 165-181, 2007.
  • L. Snyder and M. Daskin. A random-key genetic
    algorithm for the generalized traveling salesman
    problem. European Journal of Operational Research
    17 (1) 38-53, 2006.

26
Acknowledgements
  • Chris Groer, University of Maryland
  • William Mennell, University of Maryland
  • John Silberholz, University of Maryland
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