**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 >

**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)** 2 • Partition the nodes into clusters. 3 6 1 4 5

**GTSP Example (continued)** 2 • Find the minimum tour visiting each cluster. 3 6 1 4 5 Applications >

**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 >

**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 (>90 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 Objectives>

**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 (> 90 clusters). Overview >

**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 >

**Review of mrOX GA** Start Initialization Phase (Light-Weight): • Sequentially generates and evolves 7 isolated populations of 50 individuals each. • Only use the rOX for crossover. • Apply one cycle of 2-opt followed by 1-swap improvement heuristic only to new best solution. • 5% chance of mutation. • Terminate each population after no new best solution is found for 10 consecutive generations. Pop 1 Pop 2 … Pop 7 Select the best 50 individuals from the union of the isolated populations to continue on. Merge Post-Merge Phase: • Use the full mrOX for crossover. • Apply full cycles of 2-opt followed by 1-swap improvements until no improvements are found to: • Child solutions that are better than both parents. • Random 5% of population (preserve diversity). • 5% chance of mutation. • Terminate after no new best solution is found for 150 consecutive generations. Post-Merge End Crossover and Local Search >

**Review of mrOX GA** Initialization Phase: rOX Crossover and Local Search Improvements New Best? rOX Crossover no yes 2-opt 1-swap Post-Merge Phase: mrOX Crossover and Local Search Improvements Better than Parents? mrOX Crossover no yes 2-opt 1-swap until no improvements mrOX Crossover >

**Review of mrOX Crossover** Randomly Generate Cut Points Permutations of Rotations and Reversals Combinations of Cluster Nodes at End-Points P1 X Z 1 2 3 4 Y 1 2 3 4 X Y Z Z {a,b} X Y {c,d} P1’ X Z 1 2 3 4 Y Y Z X x x Z X Y Z a X Y c 1 2 3 4 Z a X Y d Z b X Y c Z Y X Z b X Y d X Y Z Y X Z X Z Y X Y Z P2’ 3 1 2 4 Combined Tour P2 3 X 1 2 Y 4 Z Z b X Y d Example >

**Example of mrOX Crossover** X X P2 Y Y Z Z P1 x x 1 4 1 4 2 3 2 3 X Y X Z Insert the best path. Yd Zb Check rotations and reversals with node combinations at the end points. X 1 4 Yd Zb 2 3 Child Complexity >

**Complexity of mrOX Crossover** The ordered set of cluster/node pairs added from the 2nd parent. The cardinality of the set Sp2. The number of comparisons required in a mrOX crossover operation. is the maximum number of nodes in the clusters. where A function that returns the number of nodes in the cluster at index i in the ordered set Sp2. Bounded by: Local Search >

**Local Search: 2-opt & 1-swap** X X Y Y Z Z 2-opt 1 4 1 4 2 3 2 3 X X X X Y Y Z 1 1 1 X 2 2 2 3 3 3 4 4 4 Z Y Z Y Y Z Z 1-swap 1 4 1 4 2 3 2 3 X Y 1 Z 2 3 4 Mutation >

**Mutation** X Y Z (1) Randomly generate cut points. 1 4 (2) Isolate the path between the cut points. 2 3 1 4 2 3 X Y Z (3) Reverse the order of the nodes. 1 4 (4) Reinsert the path. 2 3 1 4 2 3 Overview >

**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 >

**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 >

**Type 3 Parallelism in the mrOX GA** start … • Genetic algorithms are amenable to parallelism via concurrent exploration. • Cooperation between processes can be implemented to ensure diversity while maintaining intensification. P1 P2 Pn start … P1 P2 Pn P1 P2 Pn T P1 P2 Pn end P1 P2 Pn end Parallel Cooperation w/ Mesh >

**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 >

**Presentation Overview** • Background and Review • Review of mrOX GA • Parallelism in the mrOX GA • Status Summary and Future Work Status Summary >

**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 >

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

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**Acknowledgements** Chris Groer, University of Maryland William Mennell, University of Maryland John Silberholz, University of Maryland