# Exact Weight Subgraphs and the k-Sum Conjecture

arXiv: https://arxiv.org/abs/1304.7558 · HTML: https://mlsanity.com/paper/1304.7558/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2013.

## Built on by (most cited first, 2 of 2)

- [Symmetric Sparse Boolean Matrix Factorization and Applications](https://mlsanity.com/paper/2102.01570/) (2021)
- [Hardness of High-Dimensional Linear Classification](https://mlsanity.com/paper/2603.19061/) (2026)

## Abstract (arXiv)

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n^2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star.