On Residual Approximation in Solution Extension Problems

Weller, Mathias; Château, Annie; Giroudeau, Rodolphe; König, Jean-Claude; Pollet, Valentin

doi:10.1007/978-3-319-48749-6_34

Cited by 4 publications

(8 citation statements)

References 18 publications

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“…Condition (i) of Definition 2 refers to the discussion in [22] which argues that the "residue" part of a feasible solution S, i.e., the part given in S \ U, is the most important of the valuation. Another consequence of conditions (i) and (ii) concerns the valuation of w restricted to the subgraph induced by V (U) (except for edges of U): this function does not satisfy any specified property.…”

Section: Definition 1 (C-relaxed Triangle Inequality)mentioning

confidence: 99%

“…As indicated in introduction, extending a partial solution into a feasible solution has been studied from a computational complexity for independent dominating set 5 , conference programs and coloration in [9,11,21] respectively. Dealing with approximation algorithms with performance guarantee of NP-hard of optimization problems, results on extension problems are given in [9,22] for several problems including vertex cover, connected vertex cover feedback vertex set, Steiner tree, max leaf and bin packing. For algorithms finding an optimal solution, it often does not matter whether we optimize the weight of whole solution S or the weight of the residue part S \ U .…”

Section: Related Workmentioning

confidence: 99%

“…However, in the context of approximation algorithms, this difference may produce important modifications as for bin packing. It is the main reason explaining the works given in [22] where the authors define and propose the approx-imation classes FRAPX and RAPX capturing approximability of the residue. The residue approximation of a solution S is the approximation of w(S) − w(U).…”

Section: Related Workmentioning

confidence: 99%

“…The residue approximation of a solution S is the approximation of w(S) − w(U). In the conclusion of their paper [22], the authors elaborate that obtaining parameterized complexity results [10] of extension problems parameterized by |U| leads to challenging open problems.…”

Section: Related Workmentioning

confidence: 99%

“…This corresponds to requiring some arcs to belong to any feasible collection of spanning star branchings. Motivated by scheduling or control networking applications [9,11,21,22], this type of problem, which consists of extending partial solutions, has recently drawn attention of the research community.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Complexity and approximability of extended Spanning Star Forest problems in general and complete graphs

Khoshkhah

Ghadikolaei

Monnot

et al. 2019

Theoretical Computer Science

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A solution extension problem consist in, an instance and a partial feasible solution which is given in advance and the goal is to extend this partial solution to a feasible one. Many well-known problems likes Coloring Extension Problems, Travelling Salesman Problem and Routing problems, have been studied in this framework. In this paper, we focus on the edge-weighted spanning star forest problem for both minimization and maximization versions. The goal here is to find a vertex partition of an edge-weighted graph into disjoint non-trivial stars and the value of a solution is given by the sum of the edge-weights of the stars. We propose NP-hardness, parameterized complexity, positive and negative approximation results.

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Section: Definition 1 (C-relaxed Triangle Inequality)mentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Complexity and approximability of extended Spanning Star Forest problems in general and complete graphs

Khoshkhah

Ghadikolaei

Monnot

et al. 2019

Theoretical Computer Science

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On the Approximation Hardness of Geodetic Set and Its Variants

Davot

Isenmann

Thiebaut

2021

Lecture Notes in Computer Science

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Given a graph, a geodetic set (resp. edge geodetic set) is a subset of vertices such that every vertex (resp. edge) of the graph is on a shortest path between two vertices of the subset. A strong geodetic set is a subset S of vertices and a choice of a shortest path for every pair of vertices of S such that every vertex is on one of these shortest paths. The geodetic number (resp. edge geodetic number) of a graph is the minimum size of a geodetic set (resp. edge geodetic set) and the strong geodetic number is the minimum size of a strong geodetic set. We first prove that, given a subset of vertices, it is N P-hard to determine whether it is a strong geodesic set. Therefore, it seems natural to study the problem of maximizing the number of covered vertices by a choice of a shortest path for every pair of a provided subset of vertices. We provide a tight 2approximation algorithm to solve this problem. Then, we show that there is no 781 /780 polynomial-time approximation algorithm for edge geodetic number and strong geodetic number on subcubic bipartite graphs with arbitrarily high girth. We also prove that geodetic number and edge geodetic number are both LOG-APX -hard, even on subcubic bipartite graphs with arbitrarily high girth. Finally, we disprove a conjecture of Iršič and Konvalinka by proving that the strong geodetic number can be computed in polynomial time in complete multipartite graphs.

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