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# Approximating CSPs with Outliers

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## Acknowledgements

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

## Cite As

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

## 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.

## Subject Classification

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

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## References

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