Variable Formulation Search for the Cutwidth Minimization Problem

Many optimization problems are formulated as min–max problems where the objective function consist of minimizing a maximum value. In this case, it is usual that many solutions of the problem has associated the same value of the objective function. When this happens it is difficult to determine which solution is more promising to continue the search. In this paper we propose a new variant of the Variable Neighbourhood Search methodology to tackle this kind of problems. The new variant, named Variable Formulation Search, makes use of alternative formulations of the problem to determine which solution is more promising when they have the same value of the objective function in the original formulation. We do that in shaking, local search and neighbourhood change steps of the basic Variable Neighbourhood Search. We apply the new methodology to the Cutwidth Minimization Problem. Computational results show that our proposal outperforms previous algorithms in the state of the art in terms of quality and computing time.

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► We propose a new variant of the VNS methodology, that we call Variable Formulation Search (VFS).
► We apply the new VFS for solving the Cutwidth Minimization Problem (CMP), to illustrate its effectiveness.
► We propose two new combinatorial formulations of the CMP.
► We derive some properties of the CMP solutions in order to reduce the size of the neighbourhoods.
► Our method compares favourable with the current state of the art.