Reoptimization of Path Vertex Cover Problem
Mehul Kumar,Amit Kumar,C. Pandu Rangan +2 more
- pp 363-374
TLDR
In this paper, the authors studied the reoptimization of path vertex cover problem and provided an algorithm with an approximation factor of 1.5, which is the best known approximation algorithm for the problem.Abstract:
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 \(I_O\), can we construct an optimal solution for the new instance \(I_N\), where \(I_N\) is the instance \(I_O\) 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 study 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 \((k \ge 4)\) on bounded degree graphs, which is also an NP-hard problem. Given a \(\rho \)-approximation algorithm for k-path vertex cover problem on bounded degree graphs, we show that it can be reoptimized within an approximation factor of \((2-\frac{1}{\rho })\) under constant number of vertex insertions.read more
Citations
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Book ChapterDOI
Reconfiguring k-path vertex covers
TL;DR: A complexity dichotomy is proved for \textsc{$k$-PVCR} on general graphs: on those whose maximum degree is $3$ (and even planar), the problem is $\mathtt{PSPACE}$-complete, while on thosewhose maximumdegree is $2$ (i.e., paths and cycles), the problems can be solved in polynomial time.
Posted Content
Reconfiguring k-path vertex covers
TL;DR: In this paper, the complexity of the problem of path vertex cover reconfiguration was investigated from the viewpoint of graph classes under the well-known reconfigureuration rules: TJS, TJ, and TAR.
References
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Noga Alon,Raphy Yuster,Uri Zwick +2 more
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Jianhua Tu,Wenli Zhou +1 more
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