Parameterized Algorithms for Partial Vertex Covers in Bipartite Graphs

On the fixed-parameter tractability of the partial vertex cover problem with a matching constraint in edge-weighted bipartite graphs

“…There, the authors show that this problem is FPT with respect to the vertex budget k 1 , when the vertices and edges of the bipartite graph are unweighted. In [29], by extending the result of Amini et al [1], it is shown that the decision version of WPVCB is FPT with respect to k 1 , if the vertices have cost one, while the edges may have arbitrary weights. On the other hand, the problem is W [1]-hard for arbitrary vertex weights, even when edges have profit one [29].…”

“…In [29], by extending the result of Amini et al [1], it is shown that the decision version of WPVCB is FPT with respect to k 1 , if the vertices have cost one, while the edges may have arbitrary weights. On the other hand, the problem is W [1]-hard for arbitrary vertex weights, even when edges have profit one [29]. [29] proves that for bounded-degree graphs WPVC is FPT with respect to k 1 .…”

“…On the other hand, the problem is W [1]-hard for arbitrary vertex weights, even when edges have profit one [29]. [29] proves that for bounded-degree graphs WPVC is FPT with respect to k 1 . Similar conclusion holds for WPVC with respect to k 2 .…”

“…The 2-PVCB problem studied in [29] is closely related to the constrained minimum vertex cover problem on bipartite graphs (MIN-CVCB). In the MIN-CVCB problem, we are given two parameters k A and k B , and the goal is to find a cover of the bipartite graph G = (A, B, E) (with a bipartition A and B) using at most k A vertices from A and at most k B vertices from B.…”

“…The MIN-CVCB problem is NP-complete and can be solved by a fixed parameter tractable algorithm that runs in time O(1.26 [12]. The 2-PVCB problem is a generalization of the MIN-CVCB problem in which one does not need to cover all of the edges of the input graph G. In [29], it is shown that this generalization makes the problem no longer fixed parameter tractable in k A and k B under the assumption FPT =W [1].…”