  We investigate computational problems involving large weights through the
lens of kernelization, which is a framework of polynomial-time preprocessing
aimed at compressing the instance size. Our main focus is the weighted Clique
problem, where we are given an edge-weighted graph and the goal is to detect a
clique of total weight equal to a prescribed value. We show that the weighted
variant, parameterized by the number of vertices $n$, is significantly harder
than the unweighted problem by presenting an $O(n^{3 - \varepsilon})$ lower
bound on the size of the kernel, under the assumption that NP $\not \subseteq$
coNP/poly. This lower bound is essentially tight: we show that we can reduce
the problem to the case with weights bounded by $2^{O(n)}$, which yields a
randomized kernel of $O(n^3)$ bits.
  We generalize these results to the weighted $d$-Uniform Hyperclique problem,
Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems
(CSPs). We also study weighted minimization problems and show that weight
compression is easier when we only want to preserve the collection of optimal
solutions. Namely, we show that for node-weighted Vertex Cover on bipartite
graphs it is possible to maintain the set of optimal solutions using integer
weights from the range $[1, n]$, but if we want to maintain the ordering of the
weights of all inclusion-minimal solutions, then weights as large as
$2^{\Omega(n)}$ are necessary.
