Abstract—Designing appropriate graphs is a problem frequently occurring in several common applications ranging from designing communication and transportation networks to discovering new drugs. More often than not the graphs to be designed need to satisfy multiple, sometimes conflicting, objectives e.g. total length, cost, complexity or other shape and property limitations. In this paper we present our approach to solving the multi-objective graph design problem and obtaining a set of multiple equivalent compromising solutions. Our method uses multi-objective evolutionary graphs, a graph-specific meta-heuristic optimization method that combines evolutionary algorithms with graph theory and local search techniques exploiting domain-specific knowledge. In the experimental section we present results obtained for the problem of designing molecules satisfying multiple pharmaceutically relevant objectives. The results suggest that the proposed method can provide a variety of valid solutions. Index Terms—Optimal graph design, de novo drug design, multi-objective optimization, multi-objective evolutionary algorithms MEGA. I. INTRODUCTION PTIMAL Graph Design (OGD) deals with the discovery of graph structures that satisfy certain predefined criteria while conforming to a set of specifications concerning the building blocks and the rules available for the construction of the graph. The problem is characterized by its combinatorial nature and the potentially large size of the search space defined by the number of feasible graph structures. As is the case in most real life problems, OGD is typically multi-objective, i.e. the graphs need to satisfy more than one criterion and thus, it is essentially a multi-objective combinatorial optimization problem. OGD includes a wide variety of problems including communication network design [1], route planning [2] and molecular design [3] among others. The aim of this paper is to introduce a newer development of our recently proposed multi- Manuscript received August 17, 2009. C. A. Nicolaou is with the Cyprus Institute, Nicosia, Cyprus, the University of Cyprus, Nicosia, Cyprus and Noesis Chemoinformatics, Nicosia, Cyprus. (phone: +357-22208644; fax: +357-22208625; e-mail: c.nicolaou@cyi.ac.cy). C. Kannas, is with Noesis Chemoinformatics and the University of Cyprus, Nicosia, Cyprus (e-mail: ckannas@noesisinformatics.com). C. S. Pattichis is with the Computer Science Department, University of Cyprus, Nicosia, Cyprus (e-mail: pattichi@ucy.ac.cy). objective optimization algorithm specifically designed to evolve graphs using a pool of existing building blocks and exploiting problem domain specific knowledge. The implementation of the algorithm, as well as the results presented, focus on the specific problem of molecular/drug design, also known as de novo drug design (DND). The rest of the paper is organized as follows. The next two sections of the paper briefly describe fundamental graph theory elements and their application to molecular graphs and, multi-objective optimization. Section IV of the paper introduces the proposed algorithm while section V describes the general DND problem and presents some experimental results from the application of the algorithm. The final section summarizes our conclusions and outlines some directions for future research. II. BACKGROUND A graph G = (V, E) consists of a set of vertices V(G) and a set of edges E(G). In the case of labeled graphs both vertices and edges have identifiers, i.e. each vertex and edge has a label drawn from a predefined set of vertex and edge labels. Graphs can be directed or undirected. In directed graphs edges are ordered pairs of the vertices they connect where, in undirected, edges simply list the pair of vertices they connect. Two vertices V I , V J of graph G are connected, or adjacent, if there is an edge E IJ = (V I , V J ) ∈E(G). If there is a path P = (E 1 , E 2 , …, E n ) between every pair of vertices in a graph G, then G is a connected graph. A graph S = (Vs, Es) is a subgraph of G = (V, E) if and only if V S ∈ V and E S ∈ E. If E S contains all edges in E connecting the vertices in V S then S is an induced subgraph of G. A common induced subgraph between G 1 and G 2 is a graph CS that is an induced subgraph of both G 1 and G 2 . The largest induced subgraph between G 1 and G 2 is known as the Maximum Common Induced Subgraph (MCIS). The largest contiguous common substructure is known as the Maximum Common Substructure (MCS). Optimal Graph Design Using A Knowledge-driven Multi-objective Evolutionary Graph Algorithm Christos A. Nicolaou, Christos Kannas, and Constantinos S. Pattichis, Member, IEEE O Proceedings of the 9th International Conference on Information Technology and Applications in Biomedicine, ITAB 2009 November 5-7, 2009 Larnaca, Cyprus