From payments of debts to real time distributed system reconfiguration Nouveaux défis en théorie de l'ordonnancement Luminy, may 2008 Jacques Carlier Renaud Sirdey Hervé Kérivin Dritan Nace

Outline The debts’ payment problem A PhD student in industry; Context of the work A resource constrained scheduling problem Polynomial cases A branch and bound method A simulated annealing method A branch and cut algorithm Conclusion and perspectives

The debts’ payment problem ([CAR82, Rairo]) The problem is modeled by a valued graph. The nodes of the graph are the persons and are valued by the initial capital of the persons. The arcs of the graph model the debts and are valued by their amount.

The non preemptive case (decision problem) no yes

NP-hard in the weak sense Partition problem:  a i = 2T

The preemptive case

C i : sum of claims of node x i, D i : sum of debts; Balance: B i = D i – C i ; Theorem. The problem has a solution if and only if: a) B i ≤ a i ; b) x i has an ascendant node of initial capital non null; c) x i has a descendant node of final capital non null.

The preemptive case Algorithm O(n 3 ): Compute an Eulerian cycle in a transport network obtained by adding a source and a sink;

A PhD thesis in industry Renaud Sirdey was working at the research center of NORTEL as system architect. In may 2004, he started his PhD within a convention CIFRE*. He was directed by Jacques Carlier (Heudiasyc) and Dritan Nace (Heudiasyc). His supervisor was Jacques-Olivier Bouvier (NORTEL). *CIFRE : Convention avec l’Industrie pour la Formation par la Recherche.

Operations Research at NORTEL (see [Sirdey’07, 4OR]) Load Balancing on BSC and partitioning problem Configuration of radio cells and bipartite matching Repartition of cells on MIC links and bin-packing Equity in assigning resources and theory of votes Electric consumption and max-min knapsack Dynamic allocation of resources and flows Route planification in MPLS networks Deconvolution/demodulation of GSM bursts and quadratic programming

Context of the work Simplified architecture of a GSM network

Fault tolerant systems At the starting time, BSC initial state has nice properties as equirepartition of loads These properties are lost due to successive failures So it is necessary to restore the system thanks to a final state having also nice properties

The reconfiguration problem A process can be : - moved from its current processor to its final processor without impact on the service. - interrupted and started again. Interruption permits to solve blocking situations. We have exactly the debt’s payment problem in the non preemptive case (plus interruption of payments). We have a degraded current state and a final state. Our objective is to move from the current state to the final state without violating capacities of processors.

A resource constrained scheduling problem with application to distributed system reconfiguration

Complexity, polynomial cases Transfer graph, G = (V,A) V = processors A = transfers (arcs are in the opposite direction) NP-hard in the strong sense. Polynomial cases : - homogeneous case - transfer graph without directed cycle - Decomposition property: connected components can be treated independently in the reverse order of a good numbering.

Branch and bound method node : (I, J,  J, R) root : ( , ,  , M) Branching rule : concatenate a transfer from R to  J, in respecting feasibility. Pseudo-polynomial lower bound : knapsack problem. Dominance rule : eliminate equivalent schedules.

Numerical results

A simulated annealing method (I) Simulated annealing : convergence to a steady state A solution  of a minimization problem is ( ,  ) feasible if Let e 1 ≤ z ≤ e P, the solution values of Metropol’s algorithm at temperature are ( ,  ) feasible.

A simulated annealing method (II) Neighboring is based on 2-opt. so Computing of c(  ) in O(|M|) The decreasing temperature law is chosen such that: The number of steps is Algorithm complexity is

Numerical results Parameters:  = 0.95,  = 0.05,  = 0.1. With a benchmark of 1020 difficult instances: For 23 instances with  > 5%, It has been necessary to launch again the algorithm in average 1.7 times. It remains three open instances.

A branch and cut method

Numerical results Starting form a good solution thanks to the Simulated Annealing algorithm. Solving relaxation associated with capacity constraints. - vO(M 2 ) minicliques - O(M 2 ) minimonocyles - O(M 3 ) transitivity constraints - s- covering and t-covering constraints Branching

Numerical results

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