On Finding and Enumerating Maximal and Maximum k-Partite Cliques in k-Partite Graphs

Let k denote an integer greater than 2, let G denote a k-partite graph, and let S denote the set of all maximal k-partite cliques in G. Several open questions concerning the computation of S are resolved. A straightforward and highly-scalable modification to the classic recursive backtracking approach of Bron and Kerbosch is first described and shown to run in O(3(n/3)) time. A series of novel graph constructions is then used to prove that this bound is best possible in the sense that it matches an asymptotically tight upper limit on vertical bar S vertical bar. The task of identifying a vertex-maximum element of S is also considered and, in contrast with the k = 2 case, shown to be NP-hard for every k >= 3. A special class of k-partite graphs that arises in the context of functional genomics and other problem domains is studied as well and shown to be more readily solvable via a polynomial-time transformation to bipartite graphs. Applications, limitations, potentials for faster methods, heuristic approaches, and alternate formulations are also addressed.