Near-optimal linear decision trees for k-SUM and related problems [article]

Daniel M. Kane and Shachar Lovett and Shay Moran
2017 arXiv   pre-print
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 ^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
more » ... ng 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.
arXiv:1705.01720v1 fatcat:nut5sxgzybdkxpqakm2r7bg2ui