Losing Weight by Gaining Edges.

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