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.
Another popular variant of the VC problem is the (MVC) problem. Given a graph @math and a positive integer @math , the is about finding @math vertices that maximize the number of distinct edges covered by them in @math . Ageev and Sviridenko in @cite_1 gave a @math -approximation algorithm for the MVC problem. An approximation algorithm, that uses a semidefinite programming technique, based on a parameter whose factor of approximation is better than @math when the parameter is sufficiently large was shown in @cite_2 . Apollonio and Simeone in @cite_6 proved that the MVC problem on bipartite graphs is NP-hard. The same authors in @cite_10 gave a @math factor approximation algorithm for MVC on bipartite graphs that exploited the structure of the fractional optimal solutions of a linear programming formulation for the problem. The authors of @cite_14 improved this result to obtain an @math factor approximation algorithm for MVC on bipartite graphs.
@cite_14 @cite_1 @cite_6 @cite_2 @cite_10
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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. 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. Improving on the Voronoi cell based techniques of Micciancio and Voulgaris (SIAM J. Comp. 13), and Sommer, Feder and Shalvi (SIAM J. Disc. Math. 09), we give a Las Vegas @math expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the @math deterministic runtime of the Micciancio Voulgaris algorithm, henceforth MV, for CVPP (which also solves CVP) at the cost of a polynomial amount of randomness (which only affects runtime, not correctness). As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that given the Voronoi relevant vectors of a lattice - the lattice vectors inducing facets of the Voronoi cell - as preprocessing and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the @math path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors. As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest. Linear programming relaxations have been used extensively in designing approximation algorithms for optimization problems. For vertex cover, linear programming and a thresholding technique gives a 2-approximate solution, rivaling the best known performance ratio. For a generalization of vertex cover we call vc t, in which we seek to cover t edges, this technique may not yield a feasible solution at all. We introduce a new method for massaging a linear programming solution to get a good, feasible solution for vc t. Our technique manipulates the values of the linear programming solution to prepare them for thresholding. We prove that this method achieves a performance ratio of 2 for vc t with unit weights. A second algorithm extends this result, giving a 2-approximation for vc t with arbitrary weights. We show that this is tight in the sense that any α-approximation algorithm for vc t with α < 2 implies a breakthrough α-approximation algorithm for vertex cover.