Partitioning a Graph into Small Pieces with Applications to Path Transversal

Lee, Euiwoong

doi:10.1137/1.9781611974782.101

Cited by 18 publications

(14 citation statements)

References 35 publications

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“…The above theorem is based on our improved results on k-Subset Edge Separator. The previous best approximation algorithm for k-Subset Edge Separator was an O(log k)-approximation that runs in time n O(1) [Lee18]. While the existence of an O(1)-approximation algorithm that runs in time n O(1) would refute the Small Set Expansion Hypothesis [RST12], we show that one can get significantly better approximations factor using k as a parameter.…”

Section: Edge Deletion Problemsmentioning

confidence: 78%

“…E.g., an (O(1), O(1))-bicriteria approximation for k-SVS would give an O(1)-approximation for H-Vertex Deletion running in (g(n, O(t)) + f (n, O(t))) log n time. The best approximation for k-SVS currently gives an (O(log k), 2)-bicriteria approximation and runs in time n O(1) [Lee18], so Theorem 1 implies the following corollaries. All algorithms in this paper are deterministic.…”

Section: Vertex Deletion Problemsmentioning

confidence: 95%

“…Such an approximation algorithm could be used to obtain better kernels [ALM + 18, ALM + 17]. In addition to the 2-approximation algorithms for Feedback Vertex Set [BG96, BBF99,CGHW98], the systematic study of the parameterized approximability depending on F has also been done in the context of both parameterized algorithms and approximation algorithms [FJP10, FLMS12, GL15, BCKP18, JP17, KS17, BRU17, ALM + 18, Lee18,KK18].…”

Section: Introductionmentioning

confidence: 99%

“…• Specifically designed linear programming (LP) relaxations for the problem, often inspired by other classical optimization problems: These approaches were used in [FJP10, GL15,Lee18] to solve Diamond Hitting Set, k-Star Transversal and k-Path Transversal, where the underlying classical problems were Feedback Vertex Set, Dominating Set, and Balanced Separator respectively. While these tools often give principled ways to find optimal approximation ratios, the previous connections were tailored to specific settings.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Losing Treewidth by Separating Subsets

Gupta¹,

Lee²,

Li³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the problem of deleting the smallest set S of vertices (resp. edges) from a given graph G such that the induced subgraph (resp. subgraph) G \ S belongs to some class H. We consider the case where graphs in H have treewidth bounded by t, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called k-Subset Vertex Separator and k-Subset Edge Separator, respectively.For the vertex deletion setting, our framework combined with the current best result for k-Subset Vertex Separator, improves approximation ratios for basic problems such as k-Treewidth Vertex Deletion and Planar-F Vertex Deletion. Our algorithms are simpler than previous works and give the first deterministic and uniform approximation algorithms under the natural parameterization.For the edge deletion setting, we give improved approximation algorithms for k-Subset Edge Separator combining ideas from LP relaxations and important separators. We present their applications in bounded-degree graphs, and also give an APX-hardness result for the edge deletion problems.3 The conference version of [Lee18] presents a randomized algorithm, but the journal version derandomized it.

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Section: Edge Deletion Problemsmentioning

confidence: 78%

Section: Vertex Deletion Problemsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Losing Treewidth by Separating Subsets

Gupta¹,

Lee²,

Li³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

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show abstract

“…However, unweighted and weighted k-path vertex cover problems on trees have polynomial time algorithms [2] [3]. The problem has been studied in [6] as k-path traversal problem which presents a log(k)-approximation algorithm for the unweighted version. For k = 2, the k-P V CP corresponds to the conventional vertex cover problem.…”

Section: Related Work and Contributionsmentioning

confidence: 99%

Reoptimization of Path Vertex Cover Problem

Kumar

Rangan

2019

Lecture Notes in Computer Science

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Most optimization problems are notoriously hard. Considerable efforts must be spent in obtaining an optimal solution to certain instances that we encounter in the real world scenarios. Often it turns out that input instances get modified locally in some small ways due to changes in the application world. The natural question here is, given an optimal solution for an old instance IO, can we construct an optimal solution for the new instance IN , where IN is the instance IO with some local modifications. Reoptimization of NP-hard optimization problem precisely addresses this concern. It turns out that for some reoptimization versions of the NP-hard problems, we may only hope to obtain an approximate solution to a new instance. In this paper, we specifically address the reoptimization of path vertex cover problem. The objective in k-path vertex cover problem is to compute a minimum subset S of the vertices in a graph G such that after removal of S from G there is no path with k vertices in the graph. We show that when a constant number of vertices are inserted, reoptimizing unweighted k-path vertex cover problem admits a PTAS. For weighted 3-path vertex cover problem, we show that when a constant number of vertices are inserted, the reoptimization algorithm achieves an approximation factor of 1.5, hence an improvement from known 2-approximation algorithm for the optimization version. We provide reoptimization algorithm for weighted k-path vertex cover problem (k ≥ 4) on bounded degree graphs, which is also an NP-hard problem. Given a ρ-approximation algorithm for k-path vertex cover problem on bounded degree graphs, we show that it can be reoptimized within an approximation factor of (2 − 1 ρ ) under constant number of vertex insertions.

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