Paths to Trees and Cacti

Agrawal, Akanksha; Kanesh, Lawqueen; Saurabh, Saket; Tale, Prafullkumar

doi:10.1007/978-3-319-57586-5_4

Cited by 7 publications

(12 citation statements)

References 28 publications

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“…See Figure 1 for an example. We note that the number of different types is at 1) . We need following an auxiliary function corresponding to a matching, which will be useful in defining the type for a matching.…”

Section: Definition 2 (Typementioning

confidence: 98%

“…The goal of parameterized complexity is to find ways of solving NP-hard problems more efficiently than brute force by associating a small parameter to each instance. Formally, a parameterization of a problem is assigning a positive integer parameter k to each input instance and we say that a parameterized problem is fixed-parameter tractable (FPT) if there is an algorithm, that given an instance (I, k), resolves in time bounded by f (k)•|I| O (1) , where |I| is the size of the input I and f is an arbitrary computable function depending only on the parameter k.…”

Section: Proposition 1 ([22] Vizing) For Any Simple Graphmentioning

confidence: 99%

“…, |X |} for which S j = N (w). We remark that the values of all the functions defined above can be computed in (total) time bounded by 2 O(|X| 2 ) • n O (1) .…”

Section: Definition 2 (Typementioning

confidence: 99%

“…Consider Case (1). Since the algorithm failed for the first time, Γ (S i ) is an H-degree balanced set and each vertex in it has a degree at most p, before the algorithm started processing for the iteration for α and T. As Γ (S i ) at the current processing contains a vertex of degree p in H, every vertex in it has degree either (p − 1) or p in H. Suppose there are n 0 vertices in Γ (S) which have degree (p − 1).…”

Section: T∈t;α∈[p0]mentioning

confidence: 99%

“…We prove that there exists an algorithm which given a graph G on n vertices and integers l, p as input either outputs a subgraph H of G such that H is p-edge colorable and has at least l edges, or correctly concludes that no such subgraph exists. Moreover, the algorithm terminates in time f (vc(G)) • n O (1) , where f (vc(G)) is some computable function which depends only on vc(G).…”

Section: T∈t;α∈[p0]mentioning

confidence: 99%

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Parameterized Complexity of Maximum Edge Colorable Subgraph

Agrawal¹,

Kundu²,

Sahu³

et al. 2020

Preprint

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A graph H is p-edge colorable if there is a coloring ψ : E(H) → {1, 2, . . . , p}, such that for distinct uv, vw ∈ E(H), we have ψ(uv) = ψ(vw). The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that |E(H)| ≥ l. We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p + k, where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) l − mm(G), where mm(G) is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with O(k • p) vertices. Furthermore, we show that there is no kernel of size O(k 1− • f (p)), for any > 0 and computable function f , unless NP ⊆ coNP/poly.

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Section: Definition 2 (Typementioning

confidence: 98%

Section: Proposition 1 ([22] Vizing) For Any Simple Graphmentioning

confidence: 99%

“…, |X |} for which S j = N (w). We remark that the values of all the functions defined above can be computed in (total) time bounded by 2 O(|X| 2 ) • n O (1) .…”

Section: Definition 2 (Typementioning

confidence: 99%

Section: T∈t;α∈[p0]mentioning

confidence: 99%

Section: T∈t;α∈[p0]mentioning

confidence: 99%

See 3 more Smart Citations

Parameterized Complexity of Maximum Edge Colorable Subgraph

Agrawal¹,

Kundu²,

Sahu³

et al. 2020

Preprint

Self Cite

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On the Parameterized Complexity of Contraction to Generalization of Trees

2018

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For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T be the family of graphs such that each graph in T can be made into a tree by deleting at most edges. Thus, the problem we study is T -Contraction. We design an FPT algorithm for T -Contraction running in time O((2 √ + 2) O(k+ ) · n O(1) ). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T -Contraction of size O([k(k + 2 )] ( α α−1 +1) ).

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On the Parameterized Complexity of Maximum Degree Contraction Problem

Saurabh

Tale

2022

Algorithmica

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In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n O(k) . As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH).Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time 1) , and remains NP-hard for every constant d ≥ 2 (Acta Informatica ( 2014)). We present a different FPT algorithm that runs in time 2 O(dk) • n O(1) . In particular, our algorithm runs in time 2 O(k) • n O(1) , for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly.

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Paths to Trees and Cacti

Cited by 7 publications

References 28 publications

Parameterized Complexity of Maximum Edge Colorable Subgraph

Parameterized Complexity of Maximum Edge Colorable Subgraph

On the Parameterized Complexity of Contraction to Generalization of Trees

On the Parameterized Complexity of Maximum Degree Contraction Problem

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