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# On Partial Vertex Cover and Budgeted Maximum Coverage Problems in Bipartite Graphs

Abstract : Graphs are often used to model risk management in various systems. Particularly, Caskurlu et al. 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 $\frac{8}{9}$-approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.
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https://hal.inria.fr/hal-01402014
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Bugra Caskurlu, Vahan Mkrtchyan, Ojas Parekh, K. Subramani. On Partial Vertex Cover and Budgeted Maximum Coverage Problems in Bipartite Graphs. 8th IFIP International Conference on Theoretical Computer Science (TCS), Sep 2014, Rome, Italy. pp.13-26, ⟨10.1007/978-3-662-44602-7_2⟩. ⟨hal-01402014⟩

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