  We show that the uniform Constraint Satisfaction Problem (CSP) parameterized
by the size of the solution is in W[1] (the problem is W[1]-hard and it is easy
to place it in W[3]). 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].
