Approximating independent sets in sparse graphs

Bansal, Nikhil

doi:10.1137/1.9781611973730.1

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…Halldorsson and Radhakrishnan [26] provide another local search approach based on [7] and obtain a ratio of 7 5 in linear time, and a ( 4 3 + )-ratio in time O(ne 1/ ). Halldórsson and Yoshihara [28] present a linear time greedy algorithm with an approximation ratio of 3 2 . 2 For the minimum vertex cover (MVC) problem in general, Garey and Johnson [20] presented a 2approximation algorithm on general graphs.…”

Section: Further Related Workmentioning

confidence: 99%

Ultimate greedy approximation of independent sets in subcubic graphs

Krysta

Mari²,

Zhi

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. It is known for its inherent hardness of approximation.We focus on the well known minimum degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been very widely studied, where it is augmented with an advice that tells the greedy which minimum degree vertex to choose if it is not unique.Our main contribution is a new mathematical theory for the design of such greedy algorithms with efficiently computable advice and for the analysis of their approximation ratios. With this new theory we obtain the ultimate approximation ratio of 5/4 for greedy on graphs with maximum degree 3, which completely solves the open problem from the paper by Halldórsson and Yoshihara (1995). Our algorithm is the fastest currently known algorithm with this approximation ratio on such graphs. We also obtain a simple and short proof of the (D+2)/3-approximation ratio of any greedy on graphs with maximum degree D, the result proved previously by Halldórsson and Radhakrishnan (1994). We almost match this ratio by showing a lower bound of (D+1)/3 on the ratio of any greedy algorithm that can use any advice. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 6/5-approximation algorithm than the one currently known.We complement our positive, upper bound results with negative, lower bound results which prove that the problem of designing good advice for greedy is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on such previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of greedy advice is non-trivial.

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Section: Further Related Workmentioning

confidence: 99%

Ultimate greedy approximation of independent sets in subcubic graphs

Krysta

Mari²,

Zhi

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In the sequential setting, an excellent summary of the known results is given by [7], which we overview in what follows. For general graphs, the best known algorithm achieves an O(n log 2 log n/ log 3 n)-approximation factor [23].…”

Section: Sequential Algorithmsmentioning

confidence: 99%

Improved Distributed Approximations for Maximum Independent Set

Kawarabayashi¹,

Khoury²,

Schild³

et al. 2019

Preprint

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We present improved results for approximating Maximum Independent Set (MaxIS) in the standard LOCAL and CONGEST models of distributed computing. Let n and ∆ be the number of nodes and maximum degree in the input graph, respectively. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a ∆-approximation to MaxIS in O(MIS(n, ∆) log W ) rounds, where MIS(n, ∆) is the running time for finding a maximal independent set, and W is the maximum weight of a node in the network. Whether their algorithm is randomized or deterministic depends on the MIS algorithm that they use as a black-box. Our results:1. Weighted Graphs: A deterministic O(MIS(n, ∆)) rounds algorithm for O(∆)-approximation to MaxIS in the CONGEST model. Weighted Graphs:A randomized 2 O( √ log log n) rounds algorithm that finds, with high probability, an O(∆)-approximation to MaxIS in the CONGEST model.

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“…In this section, we briefly illustrate how to make Bansal's argument about the integrality gap of the lifted SDP [Ban15] algorithmic. Consider the SA + (d) relaxation on G, and let sdp(G) denote its value.…”

Section: An Algorithm Using Lift-and-projectmentioning

confidence: 99%

“…Recently, Bansal [Ban15] leveraged some of these ideas to improve the approximation guarantee by a modest O(log log d) factor to d/ log d using polylog(d) levels of the SA + hierarchy. His improvement was based on combining properties of the SA + hierarchies together with the ideas of [AEKS81].…”

Section: Introductionmentioning

confidence: 99%

On the Lovász Theta Function for Independent Sets in Sparse Graphs

Bansal¹,

Gupta²,

Guruganesh³

2018

SIAM J. Comput.

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We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the integrality gap of the Lovász ϑ-function based SDP isThis improves on the previous best result of O(d/ log d), and almost matches the integrality gap of O(d/ log 2 d) recently shown for stronger SDPs, namely those obtained using poly log(d) levels of the SA + semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of K r -free graphs for large values of r.We also show how to obtain an algorithmic version of the above-mentioned SA + -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of O(d/ log 2 d) matches the best unique-games-based hardness result up to lowerorder poly(log log d) factors.

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Approximating independent sets in sparse graphs

Abstract: We consider the maximum independent set problem on sparse graphs with maximum degree d.

Cited by 10 publications

References 15 publications

Ultimate greedy approximation of independent sets in subcubic graphs

Ultimate greedy approximation of independent sets in subcubic graphs

Improved Distributed Approximations for Maximum Independent Set

On the Lovász Theta Function for Independent Sets in Sparse Graphs

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