Given a graph @math , a non-negative integer @math , and a weight function that maps each vertex in @math to a positive real number, the is about finding a maximum weighted independent set in @math of cardinality at most @math . A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in @math , is called the problem. In other words, the MIVC problem is about finding an independent set of cardinality at most @math with maximum coverage. Since it is a generalization of the well-known Maximum Weighted Independent Set (MWIS) problem, MWBIS too does not have any constant factor polynomial time approximation algorithm assuming @math . In this paper, we study MWBIS in the context of bipartite graphs. We show that, unlike MWIS, the MIVC (and thereby the MWBIS) problem in bipartite graphs is NP-hard. Then, we show that the MWBIS problem admits a @math -factor approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.
Graphs are often used to model risk management in various systems. Particularly, in [6] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. It is well-known that the vertex cover problem is in P on bipartite graphs; however, the computational complexity of the partial vertex cover problem on bipartite graphs is open. In this paper, we show that the partial vertex cover problem is NP-hard on bipartite graphs. Then, we show that the budgeted maximum coverage problem (a problem related to partial vertex cover problem) admits an ( 8 9 )-approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation. In this paper we demonstrate a general method of designing constant-factor approximation algorithms for some discrete optimization problems with cardinality constraints. The core of the method is a simple deterministic ("pipage") procedure of rounding of linear relaxations. By using the method we design a (1-(1-1 k)k)-approximation algorithm for the maximum coverage problem where k is the maximum size of the subsets that are covered, and a 1 2-approximation algorithm for the maximum cut problem with given sizes of parts in the vertex set bipartition. The performance guarantee of the former improves on that of the well-known (1 - e-1)-greedy algorithm due to Cornuejols, Fisher and Nemhauser in each case of bounded k. The latter is, to the best of our knowledge, the first constant-factor algorithm for that version of the maximum cut problem. Given a simple undirected graph G and a positive integer s the Maximum Vertex Coverage Problem is the problem of finding a set U of s vertices of G such that the number of edges having at least one endpoint in U is as large as possible. We prove that the Maximum Vertex Coverage problem on bipartite graphs is NP-hard and discuss several consequences related to known combinatorial optimization problems. Abstract We consider the max-vertex-cover (MVC) problem, i.e., find k vertices from an undirected and edge-weighted graph G =( V , E ), where | V |= n ⩾ k , such that the total edge weight covered by the k vertices is maximized. There is a 3 4-approximation algorithm for MVC, based on a linear programming relaxation. We show that the guaranteed ratio can be improved by a simple greedy algorithm for k >(3 4) n , and a simple randomized algorithm for k >(1 2) n . Furthermore, we study a semidefinite programming (SDP) relaxation based approximation algorithms for MVC. We show that, for a range of k , our SDP-based algorithm achieves the best performance guarantee among the four types of algorithms mentioned in this paper. Given a simple undirected graph @math and a positive integer @math , the maximum vertex coverage problem (MVC) is the problem of finding a set @math of @math vertices of @math such that the number of edges having at least one endpoint in @math is as large as possible. The problem is NP-hard even in bipartite graphs, as shown in two recent papers [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37--48; G. Joret and A. Vetta, Reducing the Rank of a Matroid, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. By exploiting the structure of the fractional optimal solutions of a linear programming formulation for the maximum coverage problem, we provide a @math -approximation algorithm for the problem. The algorithm immediately extends to the weighted version of MVC.