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

TL;DR

This paper investigates the NP-Completeness of the constrained minimum vertex cover problem in k-partite graphs, proposes an approximation algorithm, and analyzes its computational complexity.

Contribution

It establishes the NP-Completeness of the problem and introduces an approximation algorithm with complexity analysis for k-partite graphs.

Findings

MIN CVCK is NP-Complete in k-partite graphs

An approximation algorithm is proposed for MIN CVCK

Complexity analysis of the algorithm is provided

Abstract

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. 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.