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Section: New Results

Discrete Optimization Algorithms

Efficient encoding of pseudo-boolean constraints

Participants : Olivier Bailleux [ LERSIA, University of Bourgogne ] , Yacine Boufkhad, Olivier Roussel [ CNRS CRIL, University of Artois ] .

In [4] the open question of the existence of a polynomial size CNF encoding of pseudo-Boolean (PB) constraints such that generalized arc consistency (GAC) is maintained through unit propagation (UP) is answered affirmatively. All previous encodings of PB constraints either didn't allow UP to maintain GAC, or were of exponential size in the worst case. In the above cited paper an encoding that realizes both of the desired properties is presented. From a theoretical point of view, this narrows the gap between the expressive power of clauses and the one of pseudo-Boolean constraints.

Three way decomposition of permutation problems

Participants : Dominique Fortin, Ider Tseveendorj.

Given a m×m flow matrix F and a n×n distance matrix D , the Quadratic Assignment Problem (QAP) aims at minimizing the overall energy to carry the flows among the facilities assigned to locations related by the distance matrix; using a binary assignment m×n matrix X , its formulation is minimizing:

Im21 ${QAP{(D,F)}=\mfenced o=〈 c=〉 ~F,{XDX^t}~~~{~\mtext s.t.~}X\#8712 \#120083 }$(1)

where Im22 $\#120083 $ denotes, loosely speaking, the set of permutations. In more standard notations, when m<n

Im23 $\mfenced o={  \mtable{...}$(2)

Many practical problems give rise to (QAP) ; among special cases, the Traveling Salesperson Problem (TSP) corresponds to F = I the identity matrix. In order to guess the correlation structure between constraints and objective in 0-1 programming, we devised in [2] a method to firstly sample the distribution of fractional solutions of the continuous relaxation and then use this distribution to select an effective branching rule to early detect a good solution ; in this sense, it will prune many nodes in the branching tree. Computational results on multiknapsack and the maximum clique problem prove efficiency of this adaptative approach independent of the given problem. So, it was tempting to experiment this approach to more general integer programming especially those dealing with permutation such as (QAP) [1] , [25] , [24] .


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