Separation between Estimation and Approximation

Feige, Uriel; Jozeph, Shlomo

doi:10.1145/2688073.2688101

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Though it seems likely, we cannot know for sure that such an approximation algorithm exists. Feige and Jozeph even gave a proof that under some complexity assumptions there must be problems with estimation algorithms superior to the best approximation algorithms [5].…”

Section: Resultsmentioning

confidence: 99%

On the Configuration-LP of the Restricted Assignment Problem

Jansen

Rohwedder

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the classical problem of Scheduling on Unrelated Machines. In this problem a set of jobs is to be distributed among a set of machines and the maximum load (makespan) is to be minimized. The processing time pij of a job j depends on the machine i it is assigned to. Lenstra, Shmoys and Tardos gave a polynomial time 2-approximation for this problem [8]. In this paper we focus on a prominent special case, the Restricted Assignment problem, in which pij ∈ {pj, ∞}. The configuration-LP is a linear programming relaxation for the Restricted Assignment problem. It was shown by Svensson that the multiplicative gap between integral and fractional solution, the integrality gap, is at most 2 − 1/17 ≈ 1.9412 [11]. In this paper we significantly simplify his proof and achieve a bound of 2 − 1/6 ≈ 1.8333. As a direct consequence this provides a polynomial (2 − 1/6 + )-estimation algorithm for the Restricted Assignment problem by approximating the configuration-LP. The best lower bound known for the integrality gap is 1.5 and no estimation algorithm with a guarantee better than 1.5 exists unless P = NP.

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Section: Resultsmentioning

confidence: 99%

On the Configuration-LP of the Restricted Assignment Problem

Jansen

Rohwedder

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…We will use the terminology of Feige and Jozeph [FJ15] to distinguish between estimation and approximation of optimization problems (where the goal is to find a feasible solution of optimal value). An approximation algorithm is required to output a feasible solution whose value is close to the value of an optimal solution, e.g., output a feasible matching of nearoptimal size.…”

Section: Preliminariesmentioning

confidence: 99%

Almost-Smooth Histograms and Sliding-Window Graph Algorithms

Krauthgamer¹,

Reitblat²

2019

Preprint

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We study algorithms for the sliding-window model, an important variant of the datastream model, in which the goal is to compute some function of a fixed-length suffix of the stream. We explore the smooth histogram framework of Braverman and Ostrovsky (FOCS 2007) for reducing the sliding-window model to the insertion-only streaming model, and extend it to the family of subadditive functions. Specifically, we show that if a subadditive function can be approximated in the ordinary (insertion-only) streaming model, then it could be approximated also in the sliding-window model with approximation ratio larger by factor 2 + ε and space complexity larger by factor O ε −1 log w , where w is the window size.We then consider graph streams and show that many graph problems are subadditive, including maximum matching and minimum vertex-cover, thereby deriving sliding-window algorithms for them almost for free (using known insertion-only algorithms). One concrete example is a polylog (n)-space algorithm for estimating maximum-matching size in bounded-arboricity graphs with n vertices, whose approximation ratio differs by a factor of 2 from the insertion-only algorithm of McGregor and Vorotnikova (SODA 2018).We also use the same framework to improve the analysis of a known sliding-window algorithm for approximating minimum vertex-cover in general graphs. We improve the approximation ratio 8 + ε of van Handel (MSc Thesis 2016) to 4 + ε using the same space complexity O ε (n).

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“…One can then use a result of Alon [Alo96], in turn based on an elegant entropy-based approach of Shearer [She95], to show that such graphs have non-trivially large independents sets. However, this argument is non-algorithmic; it shows that the lifted SDP has a small integrality gap, but does not give a corresponding approximation algorithm with running time sub-exponential in n. This leads to the question whether this approach can be converted into an approximation algorithm that outputs a set of size Ω(log 2 d/d) times the optimal independent set, or if there is a gap between the approximability and estimability of this problem (as recently shown for an NP problem by Feige and Jozeph [FJ14]).…”

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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Separation between Estimation and Approximation

Cited by 6 publications

References 16 publications

On the Configuration-LP of the Restricted Assignment Problem

On the Configuration-LP of the Restricted Assignment Problem

Almost-Smooth Histograms and Sliding-Window Graph Algorithms

On the Lovász Theta Function for Independent Sets in Sparse Graphs

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