Finding the Maximum Edge K-Clique over a K-Partite Graph: A Stochastic Approach using Dynamic Programming

Abstract

This study presents a theoretical development of a stochastic algorithm using the features of dynamic programming for finding maximum edge k-clique over a k partite graph. Dynamic programming technique includes breaking down a larger problem into smaller sub-problems, storing it for getting the required optimal solution and reducing the repetitive computation. The algorithm has been developed based on major four considerations which are sort, divide, compare and conquer. It has been developed focusing on the bi-clique solve first which are the sub-problems Storing the result of bi-clique or sub-graphs over the given k partite graph helps to form a maximum edge k-clique. By using the stored result of the bi-clique which are the divided problems, the optimal solution of k-clique can be reached as the solutions of bi-cliques are the subset of the ultimate formed k-clique. The new algorithm uses bi-clique solve function over two different sets of vertices forming the bi-partite graph from the given k partite graph; each time starting from the smallest set of vertices and in an ascending order considering the cardinality of different sets. Memoization enables a new form of forming a maximum edge k-clique by using the sub-solutions of bi-clique.