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

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

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

GTSP Example

- Start with a set of nodes or locations to visit.

GTSP Example (continued)

- Partition the nodes into clusters.

GTSP Example (continued)

- Find the minimum tour visiting each cluster.

Applications gt

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

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

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

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

Review of mrOX GA

Start

Pop 1

Pop 2

Pop 7

Merge

Post-Merge

End

Crossover and Local Search gt

Review of mrOX GA

mrOX Crossover gt

Review of mrOX Crossover

Example gt

Example of mrOX Crossover

Complexity gt

Complexity of mrOX Crossover

Local Search gt

Local Search 2-opt 1-swap

Mutation gt

Mutation

(1) Randomly generate cut points.

Overview gt

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

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

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

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

Presentation Overview

- Background and Review
- Review of mrOX GA
- Parallelism in the mrOX GA
- Status Summary and Future Work

Status Summary gt

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

Future Work

- Obtain a user account on the Deepthought cluster.
- Continue with design and coding.
- Performance testing of parallel implementation.

End gt

References

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Strategies for Meta-Heuristics. Fleet Management

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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.

Acknowledgements

- Chris Groer, University of Maryland
- William Mennell, University of Maryland
- John Silberholz, University of Maryland