Finding Balanced Graph Bi-Partitions Using a Hybrid Genetic Algorithm

In th is pap er w e prop ose a hybrid g en etic a lgo­ rith m for th e graph b alanced b i-p a rtition prob lem , a chal­ len g in g N P -h ard com b in atoria l o p tim iza tio n prob lem aris­ ing in m any p ractica l ap p lica tion s. T h e hybrid character o f th e G A lies in th e ap p lica tion o f a h eu ristic p rocedu re to im prove can d id ate so lu tion s. T h e b asic idea b eh in d our h eu ristic is to id en tify and ex p lo it c lu sters, i.e ., subgraphs w ith a re la tiv e ly h igh ed ge den sity . T h e resu ltin g hybrid ge­ n etic a lgo rith m tu rn s ou t to be very effective , b o th in term s o f q u ality o f so lu tion s and run n in g t im e . On a large class o f b en chm ark fam ilies o f graphs, our hybrid g en etic a lgo­ rith m yie ld s resu lts o f equal or b e tte r q u ality th a n th ose ob ta in ed by all o th er h eu ristic a lgo rith m s we are aw are of, for com p arable run n in g tim es. I . I n t r o d u c t io n This paper introduces a hybrid genetic algorithm for finding approximate solutions of the graph balanced bi­ partition problem (BP problem). The algorithm is a com­ bination of a genetic algorithm and a local search procedure which is used for improving genetically created candidate solutions. Given an undirected graph, the BP problem consists of dividing the set of its nodes into two disjoint subsets con­ taining equal number of nodes1 in such a way th a t the num­ ber of graph edges connecting nodes belonging to different subsets (i.e., the cut size of the partition) is minimized. This combinatorial optimization problem arises in var­ ious practical applications like network partitioning, lay­ out and floor planning [7], VLSI (very large-scale integra­ tion) circuit placement [ 1 9 ] , etc. Due to its NP-hardness [11], the graph balanced bi-partition problem has been tackled by means of heuristic algorithms, which provide sub-optimal solutions of satisfactory quality in polynomial time. Various heuristic algorithms for the BP problem have been proposed (e.g., [13], [14]), also based on local search techniques like simulated annealing (e.g., [12]), tabu search (e.g., [8]) and hybrid genetic algorithms (e.g., [4], [16]). A recent summary of the approaches from the literature can be found in [9]. Most heuristics for the BP problem start from a bi­ partition of the graph, and move nodes from one side to the opposite one according to a suitable criterion for re­ ducing the cut size. This works well if the graph has a very regular structure. However, if this is not the case and there are ‘dense’ and ‘sparse’ subgraphs then one should *CWI, P.O. Box 94079, 1090 GB Am sterdam , The Netherlands t Università C a’ Foscari di Venezia, via Torino 155, 30173 MestreVenezia, Italy * University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands 1The graph is assumed to contain an even num ber of nodes; if it is not the case, a dummy node can be added. try to exploit this structure. Our local search procedure is based on the notion of clusters. Intuitively, a cluster of a graph G is a subgraph with a density2 which is signif­ icantly higher than the density of G. By the presence of many edges within a cluster it is likely th a t in a good bi­ partition the whole cluster is on one side. Based on this idea, we propose a twofold extension of a genetic algorithm, consisting of: • a procedure for finding clusters (pre-processing) and • a procedure for good emplacement of clusters (local improvement). The first procedure is used for determining the chromo­ somes representation, while the second procedure is used as operator on the chromosomes. We call the resulting hybrid algorithm ‘cluster emplacement genetic algorithm’ (CE-GA). The procedure for finding clusters is executed before the (hybrid) GA is run. This phase can be seen as pre­ processing with the purpose of identifying clusters and de­ termining the genetic representation, where a chromosome represents a bi-partition of the set of clusters. Clusters are identified experimentally, by means of a traditional node swap heuristic (NSH) for the BP problem, which is run for a large number of times on independent input bi-partitions. If the pre-processing phase indicates the absence of ‘real’ clusters, then the GA will work on nodes, which can be seen as clusters containing only one element. In this case NSH is also used as a local improvement procedure. Otherwise, if the pre-processing phase indicates the pres­ ence of ‘real’ clusters, we use a novel ‘cluster emplacement heuristic’ (CEH). A (not necessarily balanced) bi-partition of the set of clusters is considered as input of CEH and then a number of iterations is performed. In each iteration we select a cluster from the side containing most nodes, and move it to the other side. The choice of a cluster is determined by the quality of the resulting bi-partition, both in terms of cut size and balance. A quasi-balanced bi-partition of the graph is obtained by iterating until no further improvement can be made. We provide empirical evidence of the effectiveness of CE-GA by performing extensive experiments on two classes of random graphs, which are generally used as benchmark graphs for the graph balanced bi-partition problem. The rest of the paper is organized as follows. In Section II we present the NSH algorithm, and in the following section we explain the pre-processing phase in more detail. Next, in Section IV we describe the CEH heuristic. In Section V 2 Recall tha t the densitv of a graph with N nodes and |E | edges is 2 * \E \/(N * (N 1)). we introduce the CE-GA algorithm, and in Section VI we report on the results of our experiments and compare the performance of our algorithm with other ones. Finally, we conclude with a comparative discussion on our approach. II. A NODE SWAP HEURISTIC (NSH) In the majority of the heuristic algorithms for the BP problem, an initial bi-partition of the graph is considered, and nodes are moved from one side to the other according to a suitable criterion for reducing the cut size (see e.g., [8]). A specific heuristic is determined by the choice of the nodes to be moved and by the way they are moved (e.g., simultaneous exchange of nodes, or sequential move from one to the opposite side). In our approach, we use a simple node swap heuris­ tic (NSH) for the BP problem based on the well known Kernighan-Lin algorithm [13]. Given an initial balanced bi-partition of the graph, a number of iterations are performed. Each iteration swaps a pair of nodes, selected from those pairs yielding the most decrease in the cut size. The iteration process continues until one can no longer find a pair of nodes for which a swap would reduce the cut size. The running time per iteration is 0(d), where d is the maximum degree of a node. This is achieved by using a data structure similar to the one used in the FiducciaMattheyses variant [10] of the Kernighan-Lin algorithm. Note th a t this data structure is created only once, before starting the iteration process, and it requires 0(\E\) run­ ning time, where E is the set of edges of the input graph. The cut size is guaranteed to decrease at every iteration, therefore there is a maximum of \E\ iterations; however, in practice the number of iterations is much less. NSH is in itself very powerful, and can be used very effectively in combination with a genetic algorithm However, if the graph has a clustered structure then NSH may not perform satisfactorily. III. C l u s t e r i d e n t i f i c a t i o n Recall, th a t a cluster of a graph G is a subgraph with a density which is significantly higher than the density of G. Therefore, it is very likely th a t in a clustered graph a mini­ mal cut size can only be realized if clusters are not divided by the cut plane. Consequently, it is also likely th a t NSH keeps clusters on one side, because moving one node from a cluster to the other side would give a tem porary increase in cut size. Besides, if the majority of the nodes of a clus­ ter are in one side of the bi-partition, then moving one of the other cluster-nodes into th a t side is likely to give a reduction on the cut size. We exploit this property in the pre-processing phase for the purpose of identifying the clusters of a graph. We apply NSH to the graph for a large number of times, each time starting with a randomly chosen initial bi-partition, and then we count how often each edge occurs in the cut plane (that is, its nodes occur on opposite sides) of the resulting bi-partitions. If an edge never appears in the cut plane then it is likely to belong to a cluster. This condition however is rather strong since the ‘repair’ property of NSH is not perfect. Therefore we decide to use a weaker condition which requires th a t an edge appears in the cut plane less than a small percentage of times, called cluster-threshold. We call such an edge a cluster-edge. Clusters can now be defined as the components3 of the graph obtained by temporarily removing all non-clusteredges. Note th a t in this way every node of the graph be­ longs to exactly one cluster (which may consist of a sin­ gle node), hence clusters are pairwise disjoint. The pre­ processing phase applies NSH 100 times. Since NSH is veryfast, in practice the pre-processing phase accounts only for a small part of the total computation time.