Approximating CSPs with Outliers

Ghoshal, S and Louis, A (2022) Approximating CSPs with Outliers. In: 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022, 19 - 21 September 2022, Virtual, Urbana-Champaign.

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Abstract

Constraint satisfaction problems (CSPs) are ubiquitous in theoretical computer science. We study the problem of Strong-CSPs, i.e. instances where a large induced sub-instance has a satisfying assignment. More formally, given a CSP instance G(V, E, [k], {Πij}(i, j)∈E) consisting of a set of vertices V, a set of edges E, alphabet [k], a constraint Πij ⊂ [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.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright for this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Approximation algorithms; Optimization, Constraint graph; Constraint-satisfaction problems; Expansion properties; Exponentials; Problem instances; Satisfying assignments; Strong unique game; Theoretical computer science; Threshold rank; Unique games, Constraint satisfaction problems
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	31 Oct 2022 11:10
Last Modified:	31 Oct 2022 11:10
URI:	https://eprints.iisc.ac.in/id/eprint/77647

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