# Near-optimal linear decision trees for k-SUM and related problems

## TL;DR

This paper develops near-optimal linear decision trees for problems like k-SUM, SUBSET-SUM, and sumset sorting, using comparison-based queries with query complexity close to theoretical limits.

## Contribution

It introduces constructions of linear decision trees for combinatorial problems based on inference dimension, connecting machine learning concepts with discrete geometry.

## Key findings

- Constructed linear decision trees for k-SUM with O(n log^2 n) queries.
- Achieved near-optimal query complexity for SUBSET-SUM and sumset sorting.
- Utilized comparison queries with sparse coefficients for efficient decision trees.

## Abstract

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n \log^2 n)$ linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two $k$-subsets; when viewed as linear queries, comparison queries are $2k$-sparse and have only $\{-1,0,1\}$ coefficients. We give similar constructions for sorting sumsets $A+B$ and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms. Our constructions are based on the notion of "inference dimension", recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.