# Constraint Satisfaction Parameterized by Solution Size

## Abstract

*size constraint*requires that exactly $k$ variables take nonzero values. We also study a more refined version of this restriction: a

*global cardinality constraint*prescribes how many variables have to be assigned each particular value. We study the parameterized complexity of these types of CSPs where the parameter is the required number $k$ of nonzero variables. As special cases, we can obtain natural and well-studied parameterized problems such as Independent set, Vertex Cover, $d$-Hitting Set, Biclique, etc. In the case of constraint languages closed under substitution of constants, we give a complete characterization of the fixed-parameter tractable cases of CSPs with size constraints, and we show that all the remaining problems are W[1]-hard. For CSPs with cardinality constraints, we obtain a similar classification, but for some of the problems we are only able to show that they are Biclique-hard. The exact parameterized complexity of the Biclique problem is a notorious open problem, although it is believed to be W[1]-hard.

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**Submitted**: 22 June 2012

**Accepted**: 12 November 2013

**Published online**: 15 April 2014

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