This paper focuses on integer linear programs where solutions are binary matrices, and the corresponding symmetry group is the set of all column permutations. Orbitopal fixing, as introduced in [12], is a technique designed to break symmetries in the special case of partitioning (resp. packing) formulations involving matrices with exactly (resp. at most) one 1-entry in each row. The main result of this paper is to extend orbitopal fixing to the full orbitope, defined as the convex hull of binary matrices with lexicographically nonincreasing columns. We determine all the variables whose values are fixed in the intersection of an hypercube face with the full orbitope. Sub-symmetries arising in a given subset of matrices are also considered, thus leading to define the full sub-orbitope in the case of the sub-symmetric group. We propose a linear time orbitopal fixing algorithm handling both symmetries and sub-symmetries. We introduce a dynamic variant of this algorithm where the lexicographical order follows the branching decisions occurring along the B&B search. Experimental results for the Unit Commitment Problem are presented. A comparison with state-of-the-art techniques is considered to show the effectiveness of the proposed variants of the algorithm.
DefinitionsThroughout the paper, we consider an Integer Linear Program (ILP) of the form min c(x) | x ∈ X , with X ⊆ P(m, n) and c : P(m, n) → Rwhere P(m, n) is the set of m × n binary matrices. A symmetry is defined as a permutation π of the columns {1, ..., n} such that for any solution matrix x ∈ X , matrix π(x) is also solution and has same cost, i.e., π(x) ∈ X and c(x) = c(π(x)). The symmetry group G of ILP (1) is the set of all * pascale.bendotti@edf.fr † pierre.fouilhoux@lip6.fr ‡ cecile.rottner@edf.fr such permutations. Symmetry group G partitions the solution set X into orbits, i.e., two matrices are in the same orbit if there exists a permutation in G sending one to the other. Symmetries arising in ILP can impair the solution process, in particular when symmetric solutions lead to an excessively large branch and bound (B&B) search tree (see survey [19]). Symmetry detection techniques are proposed in [15,3]. Various techniques, so called symmetry-breaking techniques, are available to handle symmetries in ILP of the form (1). The general idea is, in each orbit, to pick one solution, defined as the representative, and then restrict the solution set to the set of all representatives.A technique is said to be full-symmetry-breaking (resp. partial-symmetry-breaking) if the solution set is exactly (resp. partially) restricted to the representative set. Moreover, such a technique may introduce some specific branching rules that interfere with the B&B search. This can forbid exploiting a user-defined branching rule or, even, the default solver branching settings. A symmetrybreaking technique is said to be flexible if at any node of the B&B tree, the branching rule can be derived from any linear inequality on the variables. Such a technique can be based on specific...
