IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 17, NO. 3, MARCH 1998 193 GRCA: A Hybrid Genetic Algorithm for Circuit Ratio-Cut Partitioning Thang Nguyen Bui and Byung-Ro Moon Abstract— A genetic algorithm for partitioning a hypergraph into two disjoint graphs of minimum ratio cut is presented. As the Fiduccia–Mattheyses graph partitioning heuristic turns out to be not effective when used in the context of a hybrid genetic algorithm, we propose a modification of the Fiduccia–Mattheyses heuristic for more effective and faster space search by introducing a number of novel features. We also provide a preprocessing heuristic for genetic encoding designed solely for hypergraphs which helps genetic algorithms exploit clustering information of input graphs. Supporting combinatorial arguments for the new preprocessing heuristic are also provided. Experimental results on industrial benchmarks circuits showed visible improvement over recently published algorithms with a lower growth rate of running time. Index Terms—Circuit ratio-cut partitioning, -schemata, graph partitioning, hybrid genetic algorithm, preprocessing. I. INTRODUCTION L ET be a hypergraph, where is the set of vertices with weights, and is the set of hyperedges. A hyperedge is defined to be a set of vertices. When a circuit is mapped into this graph model, cells are mapped to vertices and nets are mapped to hyperedges. A bipartition of is a partitioning of the vertex set into two disjoint sets and The total number of nets having at least one endpoint in each of and is called the size or cut size, denoted of the bipartition. For a bipartition to be practically useful, it needs to have some balance requirement between the sizes and of the two partitions. The ratio cut of a partition was suggested by Wei and Cheng [1]; it is defined to be the ratio The ratio cut gives a penalty proportional to the degree of unbalance, but gives chances to poorly balanced partitions with considerably small cut sizes. The ratio-cut problem is the problem of finding a partition with the minimum ratio cut. It is known to be NP- hard [1]. Ratio cut has been used frequently in recent works as a metric to judge the quality of partitions [1]–[4]. Among the most successful partitioning techniques are group migration, simulated annealing, and spectral meth- ods. The Kernighan–Lin algorithm (KL) [5], [6] and the Fiduccia–Mattheyses algorithm (FM) [7] are the two most ba- sic group migration heuristics. KL improves an initial solution Manuscript received January 16, 1996. This paper was recommended by Associate Editor R. H. J. M. Otten. T. N. Bui is with the Department of Computer Science, Pennsylvania State University-Harrisburg, Middletown, PA 17057 USA. B.-R. Moon is with the Department of Computer Science, Seoul National University, Seoul, Korea. Publisher Item Identifier S 0278-0070(98)03088-7. by repeatedly selecting an equal-sized vertex subset on each side and swapping them. FM is a variation of KL, and achieves a time complexity of by using carefully designed data structures which are currently used by several researchers [1], [3]. An important property of these group migration algorithms is that they are highly dependent on the initial solutions. To effectively use this property, several researchers have suggested preprocessing heuristics to provide good initial solutions [1], [3], [8]–[11]. In our experience, FM turned out to be not effective in finding a good ratio-cut partition when combined with genetic algorithms (GA’s), especially in finding poorly balanced but low-ratio-cut partitions. In this paper, we provide a modification of FM to fit with GA’s. The most extensive report on the graph-partitioning problem (particularly with strict balance requirement) using simulated annealing was done by Johnson et al. [12], and competitive results were reported. Another common approach to the graph- partitioning problem is to use spectral methods which have shown promising results [2], [11], [13]. These methods use the second smallest eigenvalue of the spectral matrix of the input graph, and the corresponding eigenvector as the main clue for clusters detection. Genetic algorithms have been used in various problems in the area of VLSI CAD [14]–[16]. Recently, several partition- ing algorithms using genetic algorithm have been reported [17]–[20]. Most of them showed moderate success. The au- thors also used genetic algorithms for the graph-partitioning problem in [21], and provided extensive testing results on 40 theoretical benchmark graphs; overall, they showed a comfortably better result than that of simulated annealing and multistart KL. Results of [21] were on the graph bisection problem, which requires that the graph be divided exactly in half. In this paper, we consider the ratio-cut problem for VLSI circuit graphs, i.e., hypergraphs. In [22], the authors suggested the technique of preprocessing heuristic for GA’s, and reported a performance improvement due to the prepro- cessing heuristics. In this paper, we adapt and extend the technique to hypergraphs and weighted graphs, and provide a new explanation for it. We tested the algorithm on the ACM/SIGDA benchmark circuits and some other circuits. Experiments with this algorithm showed visible improvement over recently published results [1], [2], [23]. Overall, this paper proposes a GA FM-modification hybrid for circuit ratio-cut partitioning problem which out- performs existing ratio-cut heuristics [1], [2], [23] with lower growth rate of running time. The rest of this paper is organized as follows. In Section II, we introduce some preliminaries 0278–0070/98$10.00  1998 IEEE