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Approximating CSPs with Outliers

Authors Suprovat Ghoshal, Anand Louis

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Constraint Satisfaction Problems
  • Strong Unique Games
  • Threshold Rank

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Abstract

Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of Strong-CSP s, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance 𝒢(V, E, [k], {Π_{ij}}_{(i,j) ∈ E}) consisting of a set of vertices V, a set of edges E, alphabet [k], a constraint Π_{ij} ⊂ [k] × [k] for each (i,j) ∈ E, the goal of this problem is to compute the largest subset S ⊆ V such that the instance induced on S has an assignment that satisfies all the constraints.
In this paper, we study approximation algorithms for UniqueGames and related problems under the Strong-CSP framework when the underlying constraint graph satisfies mild expansion properties. In particular, we show that given a StrongUniqueGames instance whose optimal solution S^* is supported on a regular low threshold rank graph, there exists an algorithm that runs in time exponential in the threshold rank, and recovers a large satisfiable sub-instance whose size is independent on the label set size and maximum degree of the graph. Our algorithm combines the techniques of Barak-Raghavendra-Steurer (FOCS'11), Guruswami-Sinop (FOCS'11) with several new ideas and runs in time exponential in the threshold rank of the optimal set. A key component of our algorithm is a new threshold rank based spectral decomposition, which is used to compute a "large" induced subgraph of "small" threshold rank; our techniques build on the work of Oveis Gharan and Rezaei (SODA'17), and could be of independent interest.

Cite As Get BibTex

Suprovat Ghoshal and Anand Louis. Approximating CSPs with Outliers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.43

Author Details

Suprovat Ghoshal
  • University of Michgan, Ann Arbor, MI, USA
Anand Louis
  • Indian Institute of Science, Bangalore, India

Funding

  • Louis, Anand: Supported in part by SERB Award ECR/2017/003296, a Pratiksha Trust Young Investigator Award, and an IUSSTF virtual center on "Polynomials as an Algorithmic Paradigm".

Acknowledgements

The authors thank the anonymous reviewers for their helpful suggestions and comments.

References

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