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 and Operators 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 and Operators Node weight:

The Proposed Approach: Concepts and Operators Node Number:

The Proposed Approach: Concepts and Operators Node range:

The Proposed Approach: Concepts and Operators Fold Operator: – deduces an interval from a list of active nodes

The Proposed Approach: Concepts and Operators Fold Operator:

The Proposed Approach: Concepts and Operators 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 – 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..

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