Partitioning a graph into small pieces with applications to path transversal

Lee, Euiwoong

doi:10.1007/s10107-018-1255-7

Cited by 36 publications

(44 citation statements)

References 32 publications

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“…There is no known connected H that admits an FPT (or even XP) algorithm achieving a k 1−δ -approximation for some δ > 0; in particular, the parameterized approximability of k-PATH PACKING is wide open. It is conceivable that k-PATH PACKING admits a parameterized o(k)-approximation algorithm, given an O(log k)-approximation algorithm for k-PATH DELETION [116] and an improved kernel for INDUCED P 3 PACKING [117].…”

Section: Subgraph Packingmentioning

confidence: 99%

“…How can we find such a set S efficiently? Note that if S = V, then S is an O(k/ε)-vertex separator of G. Lee [116] defined a generalization of k-VERTEX SEPARATOR called k-SUBSET VERTEX SEPARATOR, where the input consists of G = (V, E), S ⊆ V, k ∈ N, and the goal is to remove the smallest number of vertices so that each connected component has at most k vertices from S, and gave an O(log k)-bicriteria approximation algorithm.…”

Section: Treewidth and Planar Minor Deletionmentioning

confidence: 99%

“…(Without the UGC, they still ruled out a (k − 1 − ε)-approximation.) Among H that are not 2-vertex-connected, there is an O(log k)-approximation algorithm when H is a star (in time n O(1) ) or a path (in time f (k)n O(1) ) [115,116,264]. The algorithm for k-path follows from the result for TREEWIDTH k-DELETION, because any graph without a k-path has treewidth at most k. Whenever H is a tree with k vertices, detecting a copy of H in G with n vertices can be done in 2 O(k) n O(1) time [265], and it is open whether there is an O(log k)-approximation algorithm for H-SUBGRAPH DELETION in time f (k) • n O(1) .…”

Section: Subgraph Deletionmentioning

confidence: 99%

“…The case k = 2n/3 is called BALANCED SEPRATOR and has been actively studied in the approximation algorithms community, and the best approximation algorithm achieves O( log n)-bicriteria approximation [234]. When k is small, O(log k)-bicriteria approximation is also possible [116].…”

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confidence: 99%

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A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

et al. 2020

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Section: Subgraph Packingmentioning

confidence: 99%

Section: Treewidth and Planar Minor Deletionmentioning

confidence: 99%

Section: Subgraph Deletionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

et al. 2020

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“…This is a contradiction since neither {1, 4, 5} nor {2, 4, 6} nor any subset is in B 4. An undirected version of this problem has been called k-path vertex cover[4], vertex cover P k[23], or k-path transversal[18] and admits an O(log k)-approximation if k is bounded by a constant[18].…”

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confidence: 99%

Approximating the discrete time-cost tradeoff problem with bounded depth

2022

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We revisit the deadline version of the discrete time-cost tradeoff problem for the special case of bounded depth. Such instances occur for example in VLSI design. The depth of an instance is the number of jobs in a longest chain and is denoted by d. We prove new upper and lower bounds on the approximability. First we observe that the problem can be regarded as a special case of finding a minimum-weight vertex cover in a d-partite hypergraph. Next, we study the natural LP relaxation, which can be solved in polynomial time for fixed d and—for time-cost tradeoff instances—up to an arbitrarily small error in general. Improving on prior work of Lovász and of Aharoni, Holzman and Krivelevich, we describe a deterministic algorithm with approximation ratio slightly less than $$\frac{d}{2}$$ d 2 for minimum-weight vertex cover in d-partite hypergraphs for fixed d and given d-partition. This is tight and yields also a $$\frac{d}{2}$$ d 2 -approximation algorithm for general time-cost tradeoff instances, even if d is not fixed. We also study the inapproximability and show that no better approximation ratio than $$\frac{d+2}{4}$$ d + 2 4 is possible, assuming the Unique Games Conjecture and $$\text {P}\ne \text {NP}$$ P ≠ NP . This strengthens a result of Svensson [21], who showed that under the same assumptions no constant-factor approximation algorithm exists for general time-cost tradeoff instances (of unbounded depth). Previously, only APX-hardness was known for bounded depth.

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Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

Dallard

Milanič

Štorgel

2020

Graph-Theoretic Concepts in Computer Science

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We continue the study of (tw, ω)-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Maximum Independent Set and related problems.In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a minmax graph invariant related to tree decompositions. Bounded tree-independence number implies both (tw, ω)-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.In this paper, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a K 5 minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph K 2,q implies the existence of a tree decomposition with independence number at most 2(q − 1).These results are obtained using a variety of tools, including ℓ-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.

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Partitioning a graph into small pieces with applications to path transversal

Cited by 36 publications

References 32 publications

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Approximating the discrete time-cost tradeoff problem with bounded depth

Treewidth Versus Clique Number in Graph Classes with a Forbidden Structure

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