Solving k-SUM using few linear queries

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c < ⌊k/2⌋. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n³ log² n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in Õ(n⌊k/2⌋+8) time, and performs O(n³ log² n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n³ log² n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-à-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth Õ(n³) for the k-SUM problem.