Corrigendum to "Variable space search for graph coloring" [Discrete Appl. Math. 156 (2008) 2551-2560]

Page 2551: replace the second paragraph by: We propose in this paper a new local search algorithm, called Variable Space Search (VSS for short). It may be seen as an extension of the Formulation Space Search methodology (FSS for short) described and used in [1–4]. FSS considers several equivalent formulations of the same problem, in the sense that there is a one-to-one correspondence between the optimal solutions of twodifferent formulations. Each formulation is associatedwith a set of neighborhoods and an objective function, and the search moves from one formulation to another when it is blocked at a local optimum or at a stationary point with the current formulation. The extension of this methodology proposed in this paper makes it possible to use non-equivalent formulations. Since we do not require any correspondence between the optimal solutions of two different formulations, translators must be defined to transform a solution when moving from one formulation to another. Page 2552: replace the second paragraph of Section 2 by: In 1997, Mladenović and Hansen [5] proposed the VNS algorithm that uses several neighborhoods to better diversify the search and better escape from local optima. In 2005, Mladenovic et al. [2] extended VNS to the FSS methodology that uses several equivalent formulations of the considered problem. Each formulation is associatedwith its own set of neighborhoods and an objective function which define a search space. In addition to the use of VNS within each formulation, FSS includes mechanisms to jump from one formulation to another. We go one step farther by considering non-equivalent formulations, in the sense that we do not require a one-to-one correspondence between the optimal solutions in two different search spaces. Page 2552: replace the last paragraph of Section 2 by: The above algorithm can be modified in various ways, for example by choosing the next search space according to the quality of the solutions it provided in the past. Since VSS does not require a one-to-one correspondence between the solutions in Si and those in Sj (i 6= j), a constraint can be relaxed in one search space Si (and violations are then penalized in the objective function fi), while it can be always satisfied in another. As a consequence, a neighborhoodwhich is appropriate for a search space Si typically generates non-feasible solutions for another search space. For comparison, the use of FSS is illustrated in [2] where a circle packing problem is solved using several formulations which combine Cartesian and polar coordinates. While different formulation spaces are considered, they all contain the same set of solutions since solutions in two different formulation spaces only differ in the way they are coded.