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, n] 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 instances of k-Clique. – The maximum node-weighted k-Clique and node-weighted kdominating set problems can be reduced to n 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 time.