Exact Weight Subgraphs and the k-Sum Conjecture

Abstract. 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(n3 ) 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(n2 ). 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.

• Type: Journal
• Year: 2013
• Where: ICALP
• Authors: Amir Abboud, Kevin Lewi
• Keywords: ICALP 2013, Programming Languages