Approximating the maximum clique minor and some subgraph homeomorphism problems.
(2007) In Theoretical Computer Science 374(1-3). p.149-158- Abstract
- We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(sqrt n) source bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the... (More)
- We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(sqrt n) source bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al.
Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O(sqrt n) source bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω(n1/2−O(1/(logn)γ)) lower bound (for some constant γ, unless NP subset ZPTIME(2^(logn)^O(1)) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound.
Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/588104
- author
- Alon, Noga ; Lingas, Andrzej LU and Wahlén, Martin LU
- organization
- publishing date
- 2007
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Graph homeomorphism, Approximation, Graph minors
- in
- Theoretical Computer Science
- volume
- 374
- issue
- 1-3
- pages
- 149 - 158
- publisher
- Elsevier
- external identifiers
-
- wos:000246170000012
- scopus:33947373382
- ISSN
- 0304-3975
- DOI
- 10.1016/j.tcs.2006.12.021
- project
- VR 2005-4085
- language
- English
- LU publication?
- yes
- id
- c480fd1b-92cc-4b3d-b27f-d55ea1d2760a (old id 588104)
- date added to LUP
- 2016-04-01 15:22:52
- date last changed
- 2022-01-28 05:05:09
@article{c480fd1b-92cc-4b3d-b27f-d55ea1d2760a,
abstract = {{We consider the “minor” and “homeomorphic” analogues of the maximum clique problem, i.e., the problems of determining the largest h such that the input graph (on n vertices) has a minor isomorphic to Kh or a subgraph homeomorphic to Kh, respectively, as well as the problem of finding the corresponding subgraphs. We term them as the maximum clique minor problem and the maximum homeomorphic clique problem, respectively. We observe that a known result of Kostochka and Thomason supplies an O(sqrt n) source bound on the approximation factor for the maximum clique minor problem achievable in polynomial time. We also provide an independent proof of nearly the same approximation factor with explicit polynomial-time estimation, by exploiting the minor separator theorem of Plotkin et al.<br/><br>
<br/><br>
Next, we show that another known result of Bollobás and Thomason and of Komlós and Szemerédi provides an O(sqrt n) source bound on the approximation factor for the maximum homeomorphic clique achievable in polynomial time. On the other hand, we show an Ω(n1/2−O(1/(logn)γ)) lower bound (for some constant γ, unless NP subset ZPTIME(2^(logn)^O(1)) on the best approximation factor achievable efficiently for the maximum homeomorphic clique problem, nearly matching our upper bound.<br/><br>
<br/><br>
Finally, we derive an interesting trade-off between approximability and subexponential time for the problem of subgraph homeomorphism where the guest graph has maximum degree not exceeding three and low treewidth.}},
author = {{Alon, Noga and Lingas, Andrzej and Wahlén, Martin}},
issn = {{0304-3975}},
keywords = {{Graph homeomorphism; Approximation; Graph minors}},
language = {{eng}},
number = {{1-3}},
pages = {{149--158}},
publisher = {{Elsevier}},
series = {{Theoretical Computer Science}},
title = {{Approximating the maximum clique minor and some subgraph homeomorphism problems.}},
url = {{http://dx.doi.org/10.1016/j.tcs.2006.12.021}},
doi = {{10.1016/j.tcs.2006.12.021}},
volume = {{374}},
year = {{2007}},
}