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Title: On Bipartite Graphs Trees and Their Partial Vertex Covers.

Abstract

Graphs can be used to model risk management in various systems. Particularly, Caskurlu et al. in [7] have considered a system, which has threats, vulnerabilities and assets, and 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. One can either restricting the permissions of the users, or encapsulating the system assets. The pointed out two strategies correspond to deleting minimum number of elements corresponding to vulnerabilities and assets, such that the flow between threats and assets is reduced below the predefined 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 has remained open. In this paper, we establish that the Partial Vertex Cover problem is NP-hard on bipartite graphs, which was also recently independently demonstrated [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37–48; G. Joret and A. Vetta, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. We then identifymore » interesting special cases of bipartite graphs, for which the Partial Vertex Cover problem, the closely related Budgeted Maximum Coverage problem, and their weighted extensions can be solved in polynomial time. We also present an 8/9-approximation algorithm for the Budgeted Maximum Coverage problem in the class of bipartite graphs. We show that this matches and resolves the integrality gap of the natural LP relaxation of the problem and improves upon a recent 4/5-approximation.« less

Authors:
 [1];  [1];  [2];  [1]
  1. West Virginia Univ., Morgantown, WV (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1427215
Report Number(s):
SAND-2015-2255J
579436
DOE Contract Number:  
AC04-94AL85000
Resource Type:
Journal Article
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Caskurlu, Bugra, Mkrtchyan, Vahan, Parekh, Ojas D., and Subramani, K.. On Bipartite Graphs Trees and Their Partial Vertex Covers.. United States: N. p., 2015. Web.
Caskurlu, Bugra, Mkrtchyan, Vahan, Parekh, Ojas D., & Subramani, K.. On Bipartite Graphs Trees and Their Partial Vertex Covers.. United States.
Caskurlu, Bugra, Mkrtchyan, Vahan, Parekh, Ojas D., and Subramani, K.. Sun . "On Bipartite Graphs Trees and Their Partial Vertex Covers.". United States. https://www.osti.gov/servlets/purl/1427215.
@article{osti_1427215,
title = {On Bipartite Graphs Trees and Their Partial Vertex Covers.},
author = {Caskurlu, Bugra and Mkrtchyan, Vahan and Parekh, Ojas D. and Subramani, K.},
abstractNote = {Graphs can be used to model risk management in various systems. Particularly, Caskurlu et al. in [7] have considered a system, which has threats, vulnerabilities and assets, and 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. One can either restricting the permissions of the users, or encapsulating the system assets. The pointed out two strategies correspond to deleting minimum number of elements corresponding to vulnerabilities and assets, such that the flow between threats and assets is reduced below the predefined 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 has remained open. In this paper, we establish that the Partial Vertex Cover problem is NP-hard on bipartite graphs, which was also recently independently demonstrated [N. Apollonio and B. Simeone, Discrete Appl. Math., 165 (2014), pp. 37–48; G. Joret and A. Vetta, preprint, arXiv:1211.4853v1 [cs.DS], 2012]. We then identify interesting special cases of bipartite graphs, for which the Partial Vertex Cover problem, the closely related Budgeted Maximum Coverage problem, and their weighted extensions can be solved in polynomial time. We also present an 8/9-approximation algorithm for the Budgeted Maximum Coverage problem in the class of bipartite graphs. We show that this matches and resolves the integrality gap of the natural LP relaxation of the problem and improves upon a recent 4/5-approximation.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2015},
month = {3}
}