Losing Weight by Gaining Edges

Amir Abboud; Kevin Lewi; Ryan Williams

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arXiv · 1311.3054

Losing Weight by Gaining Edges

Amir Abboud·Kevin Lewi·Ryan Williams

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Abstract

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.

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Amir Abboud, Kevin Lewi, Ryan Williams. 2013-11-13. Losing Weight by Gaining Edges. https://arxiv.org/abs/1311.3054

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