796 MIC2005. The 6th Metaheuristics International Conference A New Evaluation Function for the MinLA Problem Eduardo Rodriguez-Tello * Jin-Kao Hao * Jose Torres-Jimenez † * LERIA, Universit´e d’Angers 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France {ertello, hao}@info.univ-angers.fr † Mathematics Department, University of Guerrero 54 Carlos E. Adame, 39650 Acapulco Guerrero, Mexico jtj@uagro.mx 1 Introduction The minimum linear arrangement problem (MinLA) was first stated by Harper in [5]. His aim was to design error-correcting codes with minimal average absolute errors on certain classes of graphs. Latter, in the 1970’s it was used as an abstract model of the placement phase in VLSI layout, where nodes of the graph represented modules and edges represented interconnections. In this case, the cost of the arrangement measures the total wire length [1]. MinLA arises also in other application areas like graph drawing, software diagram layout and job scheduling [3]. The MinLA problem can be defined formally as follows. Let G(V,E) be a finite undirected graph, where V (|V | = n) defines the set of vertices and E ⊆ V × V = {{i, j }| i, j ∈ V } is the set of edges. Given a one-to-one function ϕ : V →{1..n}, called a linear arrangement, the total edge length for G with respect to arrangement ϕ is defined by: LA(G, ϕ)= X (u,v)∈E |ϕ(u) - ϕ(v)| (1) Then the MinLA problem consists in finding an arrangement ϕ for a given G so that LA(G, ϕ) is minimized. As is the case with many graph layout problems, finding the minimum linear arrangement is NP-hard and the corresponding decision problem is NP-complete [4]. Only in very special cases it is possible to find the optimal arrangement in polynomial time (see [3] for a detailed survey). Nowadays, the best polynomial time approximation algorithm for MinLA gives a O(log n) approximation factor for general graphs [8]. However, this algorithm presents the disadvantage of having to solve a linear program with an exponential number of constraints, making it impractical for large graphs. As an indispensable alternative, several heuristic methods have been proposed. Some examples are: a binary balanced decomposition tree heuristic [2] and a multi-scale algorithm Vienna, Austria, August 22–26, 2005