Two Heuristic Algorithms for the Minimum Weighted Connected Vertex Cover Problem Under Greedy Strategy

Abstract

The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Minimum Weighted Connected Vertex Cover problem</i> (MWCVC) is to find a subset <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> ⊂ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> ) with minimum weight in a node-weighted graph <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> , such that when removing the set <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> , the inducing graph of remaining vertices holds no edges, and the graph induced from set <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> is required to be connected. This problem comes from the classical combinatorial problem in graph theory, i.e., the Vertex Cover Problem. A large number of results on algorithms for the MWCVC problem have been reported. In this paper, we proposed two heuristic algorithms, denoted as VCC and LCVCC, to find a connected vertex cover set in a general weighted graph. The time complexity of both two algorithms are less than <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ). We compare these two algorithms with two known heuristic algorithms GR and GD (proposed by Dagdeviren in 2021) on connected graphs, and draw a conclusion that both of VCC and LCVCC perform better than GR or GD. Relatively speaking, LCVCC is expected to have better performance in dense graphs than VCC.