Near-optimal linear decision trees for k-SUM and related problems
This work provides incremental improvements in algorithm design for discrete geometry and combinatorics, benefiting researchers in theoretical computer science.
The authors tackled the problem of constructing efficient linear decision trees for combinatorial problems like k-SUM, achieving a near-optimal query complexity of O(n log^2 n) using sparse comparison queries. They extended this approach to sorting sumsets and SUBSET-SUM, with results up to poly-logarithmic terms.
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.