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Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph

Authors Venkatesan Guruswami, Ameya Velingker, Santhoshini Velusamy

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Venkatesan Guruswami
Ameya Velingker
Santhoshini Velusamy

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Venkatesan Guruswami, Ameya Velingker, and Santhoshini Velusamy. Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.8

Abstract

We study the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4-th of the total number of constraints. The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA'15) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max DICUT (a special case of Max 2CSP), in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+eps)m, where m is the total number of edges, requires polynomial space. We complement this hardness result by showing that for DICUT, one can distinguish between the case in which the optimum exceeds (1/2+eps)m and the case in which it is close to m/4. We also prove that estimating the size of the maximum acyclic subgraph of a directed graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space.
Keywords
  • approximation algorithms
  • constraint satisfaction problems
  • optimization
  • hardness of approximation
  • maximum acyclic subgraph

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