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Title: 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 polynomial-time 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 pseudo-polynomial DP-based algorithm for the most general case of having weights on both edges and vertices, which is proven to be NPhard. This algorithm provides a polynomial-time 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 » in other settings as well.« less

Authors:
 [1];  [2];  [3];  [1]
  1. West Virginia Univ., Morgantown, WV (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  3. Univ. of Haifa (Israel)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1427211
Report Number(s):
SAND-2015-2253J
579434
DOE Contract Number:  
AC04-94AL85000
Resource Type:
Book
Journal Name:
SOFSEM 2017: Theory and Practice of Computer Science
Additional Journal Information:
Other Information: ISBN 978-3-319-51962-3
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:10.1007/978-3-319-51963-0_27.
Mkrtchyan, Vahan, Parekh, Ojas D., Segev, Danny, & Subramani, K. The Approximability of Partial Vertex Covers in Trees.. United States. doi:10.1007/978-3-319-51963-0_27.
Mkrtchyan, Vahan, Parekh, Ojas D., Segev, Danny, and Subramani, K. Wed . "The Approximability of Partial Vertex Covers in Trees.". United States. doi:10.1007/978-3-319-51963-0_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 polynomial-time 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 pseudo-polynomial DP-based algorithm for the most general case of having weights on both edges and vertices, which is proven to be NPhard. This algorithm provides a polynomial-time 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 = {10.1007/978-3-319-51963-0_27},
journal = {SOFSEM 2017: Theory and Practice of Computer Science},
number = ,
volume = ,
place = {United States},
year = {2017},
month = {1}
}

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Works referenced in this record:

Distributions on level-sets with applications to approximation algorithms
conference, January 2001


Using Homogeneous Weights for Approximating the Partial Cover Problem
journal, May 2001


A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses
journal, October 2007


On the hardness of approximating vertex cover
journal, January 2005


Finding kth paths and p-centers by generating and searching good data structures
journal, March 1983


Optimization, approximation, and complexity classes
journal, December 1991

  • Papadimitriou, Christos H.; Yannakakis, Mihalis
  • Journal of Computer and System Sciences, Vol. 43, Issue 3
  • DOI: 10.1016/0022-0000(91)90023-X

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journal, May 2008


Analytical models for risk-based intrusion response
journal, July 2013


Approximation of Partial Capacitated Vertex Cover
journal, January 2010

  • Bar-Yehuda, Reuven; Flysher, Guy; Mestre, Julián
  • SIAM Journal on Discrete Mathematics, Vol. 24, Issue 4
  • DOI: 10.1137/080728044

Approximation algorithms for partial covering problems
journal, October 2004