Clique Partitions , Graph Compression and Speeding - up

We rst consider the problem of partitioning the edges of a graph G into bipartite cliques such that the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NP-complete. We then prove the existence of a partition of small total order in a suuciently dense graph and devise an eecient algorithm to compute such a partition. It turns out that our algorithm exhibits a trade-oo between the total order of the partition and the running time. Next, we deene the notion of a compression of a graph G and use the result on graph partitioning to eeciently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed representation of the input graph, thereby improving the bound on their running times, particularly on dense graphs. This makes use of the trade-oo result we obtain from our partitioning algorithm. The algorithms analyzed include those for matchings, vertex connectivity, edge connectivity and shortest paths. In each case, we improve upon the running times of the best-known algorithms for these problems.