Constraint Minimum Vertex Cover in K Partite Graph, Approximation Algorithm and Complexity Analysis

Generally, a graph G, an independent set is a subset S of vertices in G such that no two vertices in S are adjacent (connected by an edge) and a vertex cover is a subset S of vertices such that each edge of G has at least one of its endpoints in S. Again, the minimum vertex cover problem is to find a vertex cover with the smallest number of vertices. Consider a k-partite graph G = (V, E) with vertex k-partition V = P1 ∪ P2 . . . ∪ Pk and the k integers are kp1, kp2, . . . , kpk. And, we want to find out whether there is a minimum vertex cover in G with at most kp1 vertices in P1 and kp2 vertices in P2 and so on or not. This study shows that the constrained minimum vertex cover problem in k-partite graph (MIN-CVCK) is NP-Complete which is an important property of k-partite graph. Many combinatorial problems on general graphs are NP-complete, but when restricted to k-partite graph with at most k vertices then many of these problems can be solved in polynomial time. This paper also illustrates an approximation algorithm for MIN-CVCK and analyzes its complexity. In future work section, we specified a number of dimensions which may be interesting for the researchers such as developing algorithm for maximum matching and polynomial algorithm for constructing k-partite graph from general graph.