NDLI: Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

Content Provider	IEEE Xplore Digital Library
Author	Saket, R.
Copyright Year	2014
Description	Author affiliation: IBM India Res. Lab., Bangalore, India (Saket, R.)
Abstract	This work revisits the PCP Verifiers used in the works of Hastad [1], Guruswami et al. [2], Holmerin [3] and Guruswami [4] for satisfiable MAX-E3-SAT and MAX-EkSET-SPLITTING, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP ⊈ DTIME(NO(log log N)), . There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)c vertices, for some constant c > 0. . There is no polynomial time algorithm that satisfies 7/8 + 1 fraction of the clauses of a satisfiable MAX-E3(log n)c SAT instance of size n, for some constant c > 0. For any fixed k ≥ 4, there is no polynomial time algorithm that finds a partition splitting (1 - $2^{-k+1})$ + 1/(log n)c fraction of the k-sets of a satisfiable MAX-Ek-SET-SPLITTING instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al. [2] and Holmerin [3]. Similarly, our inapproximability of (log $n)^{-c}$ beyond the random assignment threshold for MAX-E3-SAT and MAX-Ek-SETSPLITTING is an exponential improvement over the previous bounds proved in [1], [3] and [4]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al. [5], [6].
Starting Page	78
Ending Page	89
File Size	316239
Page Count	12
File Format	PDF
ISBN	9781479936267
ISSN	10930159
DOI	10.1109/CCC.2014.16
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-06-11
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Color Approximation methods Labeling Polynomials Markov processes Atomic measurements Partitioning algorithms CSPs Inapproximability Hypergraph Independent Set
Content Type	Text
Resource Type	Article

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