Losing Weight by Gaining Edges

TL;DR

This paper introduces a novel encoding method for weighted sums into unweighted constraints, establishing complexity results for k-SUM and related problems, and providing improved algorithms for weighted triangle detection.

Contribution

It proves the W[1]-completeness of k-SUM and shows equivalences between weighted and unweighted problems, along with a new deterministic algorithm for weighted triangle detection.

Findings

k-SUM is W[1]-complete and reducible to k-Clique.

Weighted k-Clique and k-dominating set problems are reducible to unweighted versions.

Weighted triangle detection can be done in m^1.41 time.

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.