DRL*: A Hierarchy of Strong Block-Decomposable Linear Relaxations for 0-1 MIPs

Michel Minoux 1 Hacène Ouzia 1
1 DECISION
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : In this paper we introduce DRL*, a new hierarchy of linear relaxations for 0–1 mixed integer linear programs (MIPs), based on the idea of Reformulation–Linearization, and explore its links with the Lift-and-Project (L&P) hierarchy and the Sherali–Adams (RLT) hierarchy. The relaxations of the new hierarchy are shown to be intermediate in strength between L&P and RLT relaxations, and examples are shown for which it leads to significantly stronger bounds than those obtained from Lift-and-Project relaxations. On the other hand, as opposed to the RLT relaxations, a key advantage of the DRL* relaxations is that they feature a decomposable structure when formulated in extended space, therefore lending themselves to more efficient solution algorithms by properly exploiting decomposition. Links between DRL* and both the L&P and RLT hierarchies are further explored, and those constraints which should be added to the rank d L&P relaxation (resp to the rank d RLT relaxation) to make it coincide with the rank d DRL* relaxation (resp: to the rank d RLT relaxation) are identified. Furthermore, a full characterization of those 0–1 MIPs for which the DRL* and RLT relaxations coincide is obtained. As an application, we show that both the RLT and DRL* relaxations are the same up to rank d for the problem of optimizing a pseudoboolean function of degree d over a polyhedron. We report computational results comparing the strengths of the rank 2 L&P, DRL* and RLT relaxations. Impact on possible improved efficiency in computing some bounds for the quadratic assignment problem and other directions for future research are suggested in the conclusions.
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Michel Minoux, Hacène Ouzia. DRL*: A Hierarchy of Strong Block-Decomposable Linear Relaxations for 0-1 MIPs. Discrete Applied Mathematics, Elsevier, 2010, 158 (18), pp.2031-2048. ⟨10.1016/j.dam.2010.08.020⟩. ⟨hal-01170343⟩

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