Fixed-Parameter Algorithms for Minimum Cost Edge-Connectivity Augmentation

Marx, Dániel; Végh, László A.

doi:10.1007/978-3-642-39206-1_61

Cited by 13 publications

(15 citation statements)

References 26 publications

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“…However, if the set of links is equal to V × V it is possible to solve this problem optimally [33]. More recently, this problem has been studied in the framework of Fixed-Parameter Tractability: Végh and Marx [27] proved that this problem is in FPT when parameterized by the size of the optimal solution, and later the running time of their algorithm was further improved [3].…”

Section: Related Workmentioning

confidence: 99%

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

et al. 2021

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In the k-Connectivity Augmentation Problem we are given a k-edge-connected graph and a set of additional edges called links. Our goal is to find a set of links of minimum size whose addition to the graph makes it (k + 1)-edge-connected. There is an approximation preserving reduction from the mentioned problem to the case k = 1 (a.k.a. the Tree Augmentation Problem or TAP) or k = 2 (a.k.a. the Cactus Augmentation Problem or CacAP). While several better-than-2 approximation algorithms are known for TAP, for CacAP only recently this barrier was breached (hence for k-Connectivity Augmentation in general). As a first step towards better approximation algorithms for CacAP, we consider the special case where the input cactus consists of a single cycle, the Cycle Augmentation Problem (CycAP). This apparently simple special case retains part of the hardness of the general case. In particular, we are able to show that it is APX-hard. In this paper we present a combinatorial $\left (\frac {3}{2}+\varepsilon \right )$ 3 2 + ε -approximation for CycAP, for any constant ε > 0. We also present an LP formulation with a matching integrality gap: this might be useful to address the general case of the problem.

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Section: Related Workmentioning

confidence: 99%

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

et al. 2021

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“…This is a well known reduction (e.g. see [26]). In more details, we show that any solution J can be converted into a solution of no greater cost that has no edge incident to v, and thus v can be "shortcut".…”

Section: Proof Of Lemmamentioning

confidence: 83%

On the Tree Augmentation Problem

Nutov¹

2017

Preprint

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In the Tree Augmentation problem we are given a tree T = (V, F ) and a set E ⊆ V × V of edges with positive integer costs {c e : e ∈ E}. The goal is to augment T by a minimum cost edge set J ⊆ E such that T ∪ J is 2-edge-connected. We obtain the following results.-Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the 2-approximation barrier for instances when the maximum cost M of an edge in E is bounded by a constant; his algorithm computes a 1.96418 + approximate solution in time. Using a simpler LP, we achieve ratio 12 7 + in time 2 O(M/ 2 ) poly(n). This gives ratio better than 2 for logarithmic costs, and not only for constant costs.-One of the oldest open questions for the problem is whether for unit costs (when M = 1) the standard LP-relaxation, so called Cut-LP, has integrality gap less than 2. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most 28/15 = 2 − 2/15. In addition, we will prove that another natural LP-relaxation, that is much simpler than the ones in previous work, has integrality gap at most 7/4.

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“…We will use the following lemma to shrink edge weights so that their encoding length will be polynomial in the number of vertices and edges of the graph. It is a generalization of an idea implicitly used for weight reduction in a proof of Lokshtanov et al [47] (Theorem 4.2) and shrinks weights faster and more significantly than a theorem of Frank and Tardos [31] that is frequently used in the exact kernelization of weighted problems [3,6,25,48]. We first state the lemma, and thereafter intuitively describe its application to RPP.Lemma (lossy weight reduction).…”

Section: Preliminariesmentioning

confidence: 99%

On approximate data reduction for the Rural Postman Problem: Theory and experiments

2020

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Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and weight d of edges traversed additionally to the required ones. Thus RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I ′ with 2b + O(c/ε) vertices in O(n 3) time so that any α-approximate solution for I ′ gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on widespread benchmark data sets as well as on two real snow plowing instances from Berlin. We also make first steps toward a PSAKS for the parameter c.

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Fixed-Parameter Algorithms for Minimum Cost Edge-Connectivity Augmentation

Cited by 13 publications

References 26 publications

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

On the Cycle Augmentation Problem: Hardness and Approximation Algorithms

On the Tree Augmentation Problem

On approximate data reduction for the Rural Postman Problem: Theory and experiments

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