No sublogarithmic-time approximation scheme for bipartite vertex cover

Göös, Mika; Suomela, Jukka

doi:10.1007/s00446-013-0194-z

Cited by 7 publications

(10 citation statements)

References 18 publications

(21 reference statements)

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“…They show that even for bipartite graphs of maximum degree 3, there exists a contant ε 0 > 0 such that any randomized distributed (1 + ε 0 )-approximation algorithm for the (unweighted) minimum vertex cover problem requires Ω(log n) rounds. As our last contribution, we generalize the result of [GS14] to computing (1 + ε)-approximate solutions for any sufficiently small ε > 0.…”

Section: Our Contributionsmentioning

confidence: 97%

“…For unweighted matchings, a round complexity that is completely independent of n was obtained by [BCGS17]. Interestingly, Göös and Suomela in [GS14] showed that such a result is not possible for the minimum vertex cover problem, even in the LOCAL model. They show that even for bipartite graphs of maximum degree 3, there exists a contant ε 0 > 0 such that any randomized distributed (1 + ε 0 )-approximation algorithm for the (unweighted) minimum vertex cover problem requires Ω(log n) rounds.…”

Section: Our Contributionsmentioning

confidence: 99%

“…Theorem 1.5 is obtained by a relatively simple reduction to the (1 + ε 0 )-approximation lower bound proven in [GS14] for bipartite graphs of maximum degree 3. We note that if we only require the approximation factor to hold in expectation, as discussed at the end of Section 3.1, in the LOCAL model the theorem is tight even for general graphs and even for the weighted vertex cover problem.…”

Section: Our Contributionsmentioning

confidence: 99%

“…Recall that Göös and Suomela in [GS14] showed that there exists a constant ε 0 > 0 such that computing a (1 + ε 0 )-approximation of minimum (unweighted) vertex cover in bipartite graphs of maximum degree 3 requires Ω(log n) rounds even in the LOCAL model. We next describe the highlevel idea of extending this lower bound to show that for ε ≤ ε 0 , computing a (1 + ε)-approximate vertex cover in bipartite graphs of maximum degree 3 requires Ω log n ε rounds in the LOCAL model.…”

Section: Vertex Cover Lower Boundmentioning

confidence: 99%

“…In the LOCAL model, one can use generic techniques from [GKM17,RG20] (or the techniques from this paper) to deterministically compute a (1 + ε)-approximate minimum weighted vertex cover in time poly log n ε . Further, in [GS14], it was shown that even on bipartite graphs with maximum degree 3, there exists a constant ε 0 > 0 such that computing a (1 + ε 0 )-approximate (unweighted) vertex cover requires Ω(log n) rounds, even in the LOCAL model and even when using randomization. We generalize this result and show that for computing a (1 + ε)-approximation, one requires Ω(log(n)/ε) rounds.…”

Section: Introduction and Related Workmentioning

confidence: 99%

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Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Faour¹,

Fuchs²,

Kühn³

2021

Preprint

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We provide CONGEST model algorithms for approximating the minimum weighted vertex cover and the maximum weighted matching problem. For bipartite graphs, we show that a (1 + ε)-approximate weighted vertex cover can be computed deterministically in poly log n ε rounds. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least λ • w(V ) (where w(V ) denotes the total weight of all nodes) in polylogarithmic time in the CONGEST model, one can compute a (2 − 2λ + ε)-approximate weighted vertex cover in poly log n ε rounds in the CONGEST model. Our result in particular implies that in graphs of arboricity a, one can compute a (2 − 1/a + ε)-approximate weighted vertex cover problem in poly log n ε rounds in the CONGEST model. For maximum weighted matchings, we show that a (1 − ε)-approximate solution can be computed deterministically in time 2 O(1/ε) • poly log n in the CONGEST model. We also provide a randomized algorithm that with arbitrarily good constant probability succeeds in computing a (1 − ε)-approximate weighted matching in time 2 O(1/ε) • poly log(∆W ) • log * n, where W denotes the ratio between the largest and the smallest edge weight. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17], who gave 2 O(1/ε) • log n and 2 O(1/ε) • log ∆ log log ∆ -round randomized approximations for the unweighted matching problem.Finally, we show that even in the LOCAL model and in bipartite graphs of degree ≤ 3, if ε < ε 0 for some constant ε 0 > 0, then computing a (1 + ε)-approximation for the unweighted minimum vertex cover problem requires Ω log n ε rounds. This generalizes a result of [Göös, Suomela; DISC '12], who showed that computing a (1 + ε 0 )-approximation in such graphs requires Ω(log n) rounds.

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Section: Our Contributionsmentioning

confidence: 97%

Section: Our Contributionsmentioning

confidence: 99%

Section: Our Contributionsmentioning

confidence: 99%

Section: Vertex Cover Lower Boundmentioning

confidence: 99%

Section: Introduction and Related Workmentioning

confidence: 99%

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Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Faour¹,

Fuchs²,

Kühn³

2021

Preprint

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Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

Hefetz

Kühn

Steger

2016

Lecture Notes in Computer Science

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Abstract:We show an Ω ∆ lower bound on the runtime of any deterministic distributed O ∆ 1+η -graph coloring algorithm in a weak variant of the LOCAL model.In particular, given a network graph G = (V, E), in the weak LOCAL model nodes communicate in synchronous rounds and they can use unbounded local computation. We assume that the nodes have no identifiers, but that instead, the computation starts with an initial valid vertex coloring. A node can broadcast a single message of unbounded size to its neighbors and receives the set of messages sent to it by its neighbors. That is, if two neighbors of a node v ∈ V send the same message to v, v will receive this message only a single time; without any further knowledge, v cannot know whether a received message was sent by only one or more than one neighbor.Neighborhood graphs have been essential in the proof of lower bounds for distributed coloring algorithms, e.g., [Lin92,KW06]. Our proof analyzes the recursive structure of the neighborhood graph of the respective model to devise an Ω ∆ 1 3 − η 3 lower bound on the runtime for any deterministic distributed O ∆ 1+η -graph coloring algorithm.Furthermore, we hope that the proof technique improves the understanding of neighborhood graphs in general and that it will help towards finding a lower (runtime) bound for distributed graph coloring in the standard LOCAL model. Our proof technique works for one-round algorithms in the standard LOCAL model and provides a simpler and more intuitive proof for an existing Ω(∆ 2 ) lower bound proven in [Kuh09]. This proof also extends to one-round d-defective coloring algorithms. We show that any one round d-defective color reduction algorithm in the standard LOCAL model needs Ω(∆ 2 /(d + 1) 2 ) colors if the input graph is colored with sufficiently many colors.

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A Time Hierarchy Theorem for the LOCAL Model

Chang¹,

Pettie²

2019

SIAM J. Comput.

112

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The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the classic distributed LOCAL model has been open for many years. In particular, it is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as Op1q, Oplog˚nq, Oplog nq, 2 Op ? log nq , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. Our main results are as follows:• We define an infinite set of simple coloring problems called Hierarchical 2 1 2 -Coloring. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the klevel Hierarchical 2 1 2 -Coloring problem is Θpn 1{k q, for k P Z`. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms.• Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized n op1q -time algorithm solving the LCL can be transformed into a deterministic Oplog nq-time algorithm. Together with a previous result [6], this establishes that on trees, there are no natural deterministic complexities in the ranges ωplog˚nq-oplog nq or ωplog nqn op1q .• We expose a gap in the randomized time hierarchy on general graphs. Roughly speaking, any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in OpTLLLq time, which is the complexity of the distributed Lovász local lemma problem, currently known to be Ωplog log nq and Oplog nq.Finally, we revisit Naor and Stockmeyer's characterization of Op1q-time LOCAL algorithms for LCL problems (as order-invariant w.r.t. vertex IDs) and calculate the complexity gaps that are directly implied by their proof. For n-rings we see a ωp1q-oplog˚nq complexity gap, for p ? nˆ?nq-tori an ωp1qop a log˚nq gap, and for bounded degree trees and general graphs, an ωp1q-oplogplog˚nqq complexity gap.The goal of this paper is to understand the spectrum of natural problem complexities that can exist in the LOCAL model [31,37] of distributed computation, and to quantify the value of randomness in this model. Whereas the time hierarchy of Turing machines is known 1 to be very "smooth", recent work [6,5] has exhibited strange gaps in the LOCAL complexity hierarchy. Indeed, prior to this work it was not even known if the LOCAL model could support more than a small constant number of problem complexities. Before surveying prior work in this area, let us formally define the deterministic and randomized variants of the LOCAL model, and the class of locally checkable labeling (LCL) problems, which are intuitively those graph problems that can be computed locally in nondeterministic constant time.In bo...

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No sublogarithmic-time approximation scheme for bipartite vertex cover

Cited by 7 publications

References 18 publications

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

A Time Hierarchy Theorem for the LOCAL Model

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