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We present a new way to encode weighted sums into unweighted pairwiseconstraints, obtaining the following results.  - Define the k-SUM problem to be: given n integers in [-n^2k, n^2k] are therek which sum to zero? (It is well known that the same problem over arbitraryintegers is equivalent to the above definition, by linear-time randomizedreductions.) We prove that this definition of k-SUM remains W[1]-hard, and isin fact W[1]-complete: k-SUM can be reduced to f(k) * n^o(1) instances ofk-Clique.  - The maximum node-weighted k-Clique and node-weighted k-dominating setproblems can be reduced to n^o(1) instances of the unweighted k-Clique andk-dominating set problems, respectively. This implies a strong equivalencebetween the time complexities of the node weighted problems and the unweightedproblems: any polynomial improvement on one would imply an improvement for theother.  - A triangle of weight 0 in a node weighted graph with m edges can bedeterministically found in m^1.41 time.

Losing Weight by Gaining Edges