A Grid-enabled Branch and Bound Algorithm for Solving Challenging

Combinatorial Optimization Problems

Authors: M. Mezmaz, N. Melab and E-G. Talbi

Presented by Saad Alhowaimel

Table of Content

• Introduction• The Proposed Approach: Concepts and

Operators• The Proposed Grid-enabled Parallel B&B• Experimentation

Introduction• The problem is NP-hard and complex.• The Branch and Bound reduce the computation time, but not

exploring time.• B&B components:

– branching– Bounding– selection– elimination

• Parallelism on gird:– load balancing– fault tolerance– termination detection – global information sharing

The Proposed Approach: Concepts andOperators

• Construct a permutation tree and explore it using a DFS strategy• List of active nodes.

– Explored but not yet visited– Cover all the nodes which can be potentially explored from each node

of the list.– Algorithm stops once the list of active nodes becomes empty– Used for exploration.

• Interval– Optimize communication– Check-pointing operations

• Fold operator – Deduces an interval from a list of active nodes

• Unfold operator.– Deduces a list of active nodes from an interval

The Proposed Approach: Concepts andOperators

• Node weight:

The Proposed Approach: Concepts andOperators

• Node Number:

The Proposed Approach: Concepts andOperators

• Node range:

The Proposed Approach: Concepts andOperators

• Fold Operator:– deduces an interval from a list of active nodes

The Proposed Approach: Concepts andOperators

• Fold Operator:

The Proposed Approach: Concepts andOperators

• Unfold Operator:– Deduces a list of active node form an interval.

The Proposed Grid-enabled Parallel B&B

• Farmer-Worker Approach:– Farmer

• One processor play the role of frame• Keep a copy of all unexplored intervals

– Worker• All other processors.• Each process explores an Interval

Both worker and farmer mange the best solution found so far

The Proposed Grid-enabled Parallel B&B

• Fault Tolerance:– The coordinator manages a possible failure of the

farmer by periodically saving in tow files.– INTERVALS:

• It has a copy of all the intervals.– SOLUTION:

• It has the global best solution.– The Updating process

• Let [A,B[ be an interval being explored in a B&B process, and [A’,B’[ its copy in INTERVALS

• Exploring increase A and branching decrease B’.

The Proposed Grid-enabled Parallel B&B

• Load Balance:– The coordinator assign an interval to B&B process

using two operator:• Selection operator:

– select an interval from INTERVALS• Partitioning operator:

– divide an interval [A,B[ to [A,C[ and [C,B[.– The holder process keeps[A,C[ and requesting process keeps [C,B[ – The partitioning point C depend on the power and availability of

the processors.– At the beginning C is equal to A.– Any interval that smaller than parameterized threshold is

duplicated which speed up the search in termination phase.

The Proposed Grid-enabled Parallel B&B

• Termination Detection:– At the beginning INTERVALS has only one interval.– During exploration the number of B&B process is

equal to the number of interval while its size continuously decreases.

– The INTERVALS size is the sum of all the length of its interval.

– Any empty interval (the beginning is higher than the end) is removed automatically.

– The algorithm stop once the INTERVALS is empty.

The Proposed Grid-enabled Parallel B&B

• Solution Sharing:– The solution sharing is done by using SOLUTION. – It contains the global optimal solution. – Each B&B process manages its own local optimal

solution. – The objective: is to reflect any improvement of any

local optimal solution in all the other B&B processes. • A B&B process initializes its local optimal solution by

SOLUTION.• Immediately informs the coordinator each time its local

solution is improved • Regularly reads SOLUTION to update its local optimal

solution.

Experimentation

• The permutation flow-shop problem.– One of the important problem in industrial areas.– the problem is – Where:• The position of the machines is fixed.• A machine can handle only one job at a time.

Experimentation

• The experimentation on grid .– A problems of 50 jobs on 20 machines (Ta0561)– This instance has never been solved optimally.– The best known solution of Ta056 has a cost of 3681

with (near-optimal resolution method).– It is made up of 1889 processors belonging to 9

clusters.– Three clusters belong to three different education

departments of University de Lille.– Six clusters belong Grid’5000

Experimentation • Experiment Result.

– Two runs have been performed, and the optimal solution is found with a cost equal to 3697.

– First experiment:• The upper bound set with 3681• average of processors 500 with a peak of 1245 processors.• Experiment time One month and three weeks

– Second experiment:• The upper bound set with 3680• average of processors 328 with a peak of 1195 processors.• Experiment time is 25 days.• More than 2 million check pointing operations• About 6 billion nodes were explored. • a cumulative computation time of about 22 years..

Questions?