Uniform CSP Parameterized by Solution Size is in W[1]

Abstract: We show that the uniform Constraint Satisfaction Problem (CSP) parameterized by the size of the solution is in W1. Given a single "free" element of the domain, denoted by $0$, we define the size of an assignment as the number of variables that are mapped to a value other than $0$. Named by Kolaitis and Vardi (2000), uniform CSP means that the input contains the domain and the list of tuples of each relation in the instance. Uniform CSP is polynomial time equivalent to homomorphism problem and also to evaluation of conjunctive queries on relational databases. It also has applications in artificial intelligence. We do not restrict the problem to any (finite or infinite) family of relations. Marx and Bulatov (2014) showed that Uniform CSP restricted to some finite family of relations (thus with a bound on the arity of relations) and over any finite domain is either W[1]-complete or fixed parameter tractable. We then prove that parameterized Subset Sum with weights bounded by $n^k$ is in W[1]. Abboud et al. (2014) have already proved it, but our proof is much shorter and arguably more intuitive. Lastly, we study the weighted CSP over the Boolean Domain, where each variable is assigned a weight, and given a target value, it should be decided if there is a satisfying assignment of size $k$ (the parameter) such that the weight of its $1$-variables adds up to the target value. We prove that if the weights are bounded by $n^k$, then the problem is in W[1]. Our proofs give a nondeterministic RAM program with special properties deciding the problem. First defined by Chen et al. (2005), such programs characterize W[1].