A constant-factor approximation for weighted bond cover
The WeightedF-Vertex Deletion for a class F of graphs asks, weighted graph G, for a minimum weight vertex set S such that G−S∈F. The case when F is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for WeightedF-Ver...
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| Veröffentlicht in: | Journal of computer and system sciences Jg. 149; S. 103617 |
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| Abstract | The WeightedF-Vertex Deletion for a class F of graphs asks, weighted graph G, for a minimum weight vertex set S such that G−S∈F. The case when F is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for WeightedF-Vertex Deletion. Only three cases of minor-closed F are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class F of θc-minor-free graphs, under the equivalent setting of the Weightedc-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. Besides making an important step in the quest of (dis)proving a constant-factor approximation for WeightedF-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families. |
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| AbstractList | The WeightedF-Vertex Deletion for a class F of graphs asks, weighted graph G, for a minimum weight vertex set S such that G−S∈F. The case when F is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for WeightedF-Vertex Deletion. Only three cases of minor-closed F are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class F of θc-minor-free graphs, under the equivalent setting of the Weightedc-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. Besides making an important step in the quest of (dis)proving a constant-factor approximation for WeightedF-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families. The WFD for a class $F$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in F.$ The case when $F$ is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for WFD. Only three cases of minor-closed $F$ are known to admit constant-factor approximations, namely $\text{Vertex Cover}$, $\text{Feedback Vertex Set}$ and $\text{Diamond Hitting Set}$. We study the problem for the class $F$ of $\theta_c$-minor-free graphs, under the equivalent setting of the $pumpkin$ problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leveragea structure theorem implicit in [Joret, Paul, Sau, Saurabh, and Thomassé, SIDMA'14] which states the following: any graph $G$ containing a $\theta_c$-minor-model either contains a large two-terminal $protrusion$, or contains a constant-size $\theta_c$-minor-model, or a collection of pairwise disjoint $constant-sized$ connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for WFD, our result may be useful as a template for algorithms for other minor-closed families. |
| ArticleNumber | 103617 |
| Author | Thilikos, Dimitrios M. Kim, Eun Jung Lee, Euiwoong |
| Author_xml | – sequence: 1 givenname: Eun Jung surname: Kim fullname: Kim, Eun Jung email: eunjung.kim@kaist.ac.kr organization: KAIST, Daejeon, South Korea – sequence: 2 givenname: Euiwoong surname: Lee fullname: Lee, Euiwoong email: euiwoong@umich.edu organization: University of Michigan, Ann Arbor, USA – sequence: 3 givenname: Dimitrios M. surname: Thilikos fullname: Thilikos, Dimitrios M. email: sedthilk@thilikos.info organization: LIRMM, Univ Montpellier, CNRS, Montpellier, France |
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| Keywords | Constant-factor approximation algorithms Bonds in graphs Graph modification problems Graph minors Primal-dual method |
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| Snippet | The WeightedF-Vertex Deletion for a class F of graphs asks, weighted graph G, for a minimum weight vertex set S such that G−S∈F. The case when F is... The WFD for a class $F$ of graphs asks, weighted graph $G$, for a minimum weight vertex set $S$ such that $G-S\in F.$ The case when $F$ is minor-closed and... |
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| SubjectTerms | Bonds in graphs Computer Science Constant-factor approximation algorithms Data Structures and Algorithms Discrete Mathematics Graph minors Graph modification problems Primal-dual method |
| Title | A constant-factor approximation for weighted bond cover |
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