A polynomial solution to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-fixed-endpoint path cover problem on proper interval graphs

Asdre, Katerina; Nikolopoulos, Stavros D.

doi:10.1016/j.tcs.2009.11.003

Cited by 10 publications

(2 citation statements)

References 28 publications

(33 reference statements)

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“…The complexity of 2HP on interval graphs was asked in 1993 [12] and is still open (see [24]). It is known that 2HP is polynomial-time solvable on proper interval graphs [3]. It is natural to expect that with its full generality, ALPP would be intractable on those graph classes.…”

Section: Discussionmentioning

confidence: 99%

Parameterized Complexity of $$(A,\ell )$$-Path Packing

et al. 2021

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Given a graph $$G = (V,E)$$ G = ( V , E ) , $$A \subseteq V$$ A ⊆ V , and integers k and $$\ell $$ ℓ , the $$(A,\ell )$$ ( A , ℓ ) -Path Packing problem asks to find k vertex-disjoint paths of length exactly $$\ell $$ ℓ that have endpoints in A and internal points in $$V{\setminus }A$$ V \ A . We study the parameterized complexity of this problem with parameters |A|, $$\ell $$ ℓ , k, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $$\ell \le 3$$ ℓ ≤ 3 , while it is NP-complete for constant $$\ell \ge 4$$ ℓ ≥ 4 . We also show that the problem is W[1]-hard parameterized by pathwidth$${}+|A|$$ + | A | , while it is fixed-parameter tractable parameterized by treewidth$${}+\ell $$ + ℓ . Additionally, we study a variant called Short A-Path Packing that asks to find k vertex-disjoint paths of length at most$$\ell $$ ℓ . We show that all our positive results on the exact-length version can be translated to this version and show the hardness of the cases where |A| or $$\ell $$ ℓ is a constant.

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Section: Discussionmentioning

confidence: 99%

Parameterized Complexity of $$(A,\ell )$$-Path Packing

et al. 2021

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“…In fact, to the best of our knowledge, there is no o(|V |)-approximation algorithm for the PC problem. In the literature, several alternative objective functions have been proposed and studied [3,1,18,2,19,4,12,5]. For example, Berman and Karpinski [3] tried to maximize the number of edges on the paths in a path cover, which is equal to |V | minus the number of paths, and proposed a 7/6-approximation algorithm.…”

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Covering Vertices by Long Paths

Gong,

Edgar,

Fan

et al. 2024

Algorithmica

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Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least k vertices is considered long. When k ≤ 3, the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed k ≥ 4, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when k = 4, the problem admits a 4-approximation algorithm which was presented recently. We propose the first (0.4394k + O( 1))-approximation algorithm for the general problem and an improved 2-approximation algorithm when k = 4. Both algorithms are based on local improvement, and their performance analyses are done via amortization.

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Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs

Broersma

Fiala

Golovach

et al. 2014

Journal of Graph Theory

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Abstract.We show that for all k ≤ −1 an interval graph is −(k + 1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(n + m) time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the O(n 3 ) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is kHamilton-connected can be computed in O(n + m) time.

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A polynomial solution to the k-fixed-endpoint path cover problem on proper interval graphs

Cited by 10 publications

References 28 publications

Parameterized Complexity of $$(A,\ell )$$-Path Packing

Parameterized Complexity of $$(A,\ell )$$-Path Packing

Approximation Algorithms for Covering Vertices by Long Paths

Linear‐Time Algorithms for Scattering Number and Hamilton‐Connectivity of Interval Graphs

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