The Approximability of Partial Vertex Covers in Trees.
Abstract
Motivated by applications in risk management of computational systems, we focus our attention on a special case of the partial vertex cover problem, where the underlying graph is assumed to be a tree. Here, we consider four possible versions of this setting, depending on whether vertices and edges are weighted or not. Two of these versions, where edges are assumed to be unweighted, are known to be polynomialtime solvable (Gandhi, Khuller, and Srinivasan, 2004). However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, has not been determined yet. The main contribution of this paper is to resolve these questions, by fully characterizing which variants of partial vertex cover remain intractable in trees, and which can be efficiently solved. In particular, we propose a pseudopolynomial DPbased algorithm for the most general case of having weights on both edges and vertices, which is proven to be NPhard. This algorithm provides a polynomialtime solution method when weights are limited to edges, and combined with additional scaling ideas, leads to an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be usefulmore »
 Authors:

 West Virginia Univ., Morgantown, WV (United States)
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Univ. of Haifa (Israel)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1427211
 Report Number(s):
 SAND20152253J
579434
 DOE Contract Number:
 AC0494AL85000
 Resource Type:
 Book
 Journal Name:
 SOFSEM 2017: Theory and Practice of Computer Science
 Additional Journal Information:
 Other Information: ISBN 9783319519623
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Mkrtchyan, Vahan, Parekh, Ojas D., Segev, Danny, and Subramani, K. The Approximability of Partial Vertex Covers in Trees.. United States: N. p., 2017.
Web. doi:10.1007/9783319519630_27.
Mkrtchyan, Vahan, Parekh, Ojas D., Segev, Danny, & Subramani, K. The Approximability of Partial Vertex Covers in Trees.. United States. doi:10.1007/9783319519630_27.
Mkrtchyan, Vahan, Parekh, Ojas D., Segev, Danny, and Subramani, K. Wed .
"The Approximability of Partial Vertex Covers in Trees.". United States. doi:10.1007/9783319519630_27. https://www.osti.gov/servlets/purl/1427211.
@article{osti_1427211,
title = {The Approximability of Partial Vertex Covers in Trees.},
author = {Mkrtchyan, Vahan and Parekh, Ojas D. and Segev, Danny and Subramani, K.},
abstractNote = {Motivated by applications in risk management of computational systems, we focus our attention on a special case of the partial vertex cover problem, where the underlying graph is assumed to be a tree. Here, we consider four possible versions of this setting, depending on whether vertices and edges are weighted or not. Two of these versions, where edges are assumed to be unweighted, are known to be polynomialtime solvable (Gandhi, Khuller, and Srinivasan, 2004). However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, has not been determined yet. The main contribution of this paper is to resolve these questions, by fully characterizing which variants of partial vertex cover remain intractable in trees, and which can be efficiently solved. In particular, we propose a pseudopolynomial DPbased algorithm for the most general case of having weights on both edges and vertices, which is proven to be NPhard. This algorithm provides a polynomialtime solution method when weights are limited to edges, and combined with additional scaling ideas, leads to an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.},
doi = {10.1007/9783319519630_27},
journal = {SOFSEM 2017: Theory and Practice of Computer Science},
number = ,
volume = ,
place = {United States},
year = {2017},
month = {1}
}
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