A 1.8 approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2

Even, Guy; Feldman, Jon; Kortsarz, Guy; Nutov, Zeev

doi:10.1145/1497290.1497297

Cited by 50 publications

(108 citation statements)

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“…Given an instance of BCA, we can assume that the input graph G is a tree [5,10,7]: Each bridge-connected component of G = (V, E) can be contracted into a single vertex by contracting all edges in this component, resulting in a tree. The set of links has to be adapted accordingly.…”

Section: The Bridge-connectivity Augmentation Problemmentioning

confidence: 99%

“…To this end, we apply four data reduction rules. We begin with three data reduction rules that are also used in [7,16] and whose correctness is easy to verify.…”

Section: The Bridge-connectivity Augmentation Problemmentioning

confidence: 99%

“…The next two lemmas are used to show the upper bounds on the number and the length of such paths. The first one is due to Even et al [7] and shows that there is no such degree-two vertex path between a leaf and an internal vertex of degree at least three.…”

Section: The Bridge-connectivity Augmentation Problemmentioning

confidence: 99%

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Kernelization and Complexity Results for Connectivity Augmentation Problems

Guo

Uhlmann

Lecture Notes in Computer Science

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Connectivity augmentation problems ask for adding a set of at most k edges (called links) whose insertion makes a given graph satisfy a specified connectivity property, such as bridge-connectivity or biconnectivity. A bridge-connected (biconnected) graph is a connected graph that does not possess an edge (a vertex) whose removal results in a disconnected graph. We show that, for bridge-connectivity and biconnectivity, the respective connectivity augmentation problems admit problem kernels with O(k 2 ) vertices and links. Moreover, we study partial connectivity augmentation problems, naturally generalizing connectivity augmentation problems. Here, we do not require that, after adding the edges, the entire graph should satisfy the connectivity property, but a large subgraph. In this setting, three polynomial-time solvable connectivity augmentation problems behave differently, namely, the partial bridge-connectivity augmentation problem and the partial biconnectivity augmentation problem remain polynomial-time solvable whereas the partial strong connectivity augmentation problem becomes W[2]-hard with respect to k.

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Section: The Bridge-connectivity Augmentation Problemmentioning

confidence: 99%

“…To this end, we apply four data reduction rules. We begin with three data reduction rules that are also used in [7,16] and whose correctness is easy to verify.…”

Section: The Bridge-connectivity Augmentation Problemmentioning

confidence: 99%

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Kernelization and Complexity Results for Connectivity Augmentation Problems

Guo

Uhlmann

Lecture Notes in Computer Science

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“…Note that every instance of this problem induces a weighted set system (F , S, c), where for each edge e ∈ E there is an analogous subset S e ∈ S, consisting of all vertex sets V i ∈ F covered by e. Laminar cover can be approximated by applying various techniques, most of which actually deal with the equivalent tree augmentation problem, and produce solutions whose cost is within factor 2 of optimum. We refer the reader to a short survey of these results [12,Sect. 1].…”

Section: Laminar Covermentioning

confidence: 99%

“…1]. For the unweighted case, Nagamochi [33] proposed a (1.875 + )-approximation for any fixed > 0, a ratio that was later improved to 1.8 by Even, Feldman, Kortsarz and Nutov [12].…”

Section: Laminar Covermentioning

confidence: 99%

A Unified Approach to Approximating Partial Covering Problems

Könemann

Parekh

Segev³

2009

Algorithmica

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An instance of the generalized partial cover problem consists of a ground set U and a family of subsets S ⊆ 2 U . Each element e ∈ U is associated with a profit p(e), whereas each subset S ∈ S has a cost c(S). The objective is to find a minimum cost subcollection S ⊆ S such that the combined profit of the elements covered by S is at least P , a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈ U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection S ⊆ S that minimizes the cost of S plus the penalties of uncovered elements.Although problem-specific connections between the partial cover and the prizecollecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our An extended abstract of this paper appeared in method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.

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