# Hypergraphs and decision trees

## Abstract

We survey some recent results on a space decomposition problem that has a close relationship to the size of algebraic decision trees. These results are obtained by establishing connections between the decomposition problem and some extremal questions in hypergraphs.

Let *d, n* ≥ 1 be integers. A *d-elementary cell* is a set *D*\(\subseteq\)*R*^{ n } defined by a finite set of constraints *f*_{ i }(*x*)=0, *g*_{ j }(*x*) > 0 where *f*_{ i }, *g*_{ j } are polynomials of degrees not exceeding *d*. For any set *S*\(\subseteq\)*R*^{ n }, let *κ*_{ d }(*S*) be the smallest number of disjoint *d*-elementary cells that *S* can be decomposed into. It is easy to see that any degree-*d* (ternary) algebraic decision tree for solving the membership question of *S* must have size no less than *κ*_{ d }(*S*). Thus, any lower bound to *κ*_{ d }(*S*) yields also a lower bound the size complexity for the corresponding membership problem.

Let *A*_{ n }={(x_{1},x_{2},...,x_{n}) ¦ *x*_{ i } ≥ 0}. A well-known result of Rabin states that any algebraic decision tree for the membership question of *A*_{ n } must have height at least *n*. In this talk we discuss a recent result by Grigoriev, Karpinski and Yao [GKY], which gives an exponential lower bound to *κ*_{ d }(*A*_{ n }) for any fixed *d*, and hence to the size of any fixed degree (ternary) algebraic decision tree for solving this problem. The proof utilizes a new connection between *κ*_{ d }(*A*_{ n }) and the maximum number of minimal cutsets for any rank-*d* hypergraph on *n* vertices. We also discuss an improved lower bound by Wigderson and Yao [WY]. Open questions are presented.

## References

- [GKY]D. Grigoriev, M. Karpinski and A. Yao, “An exponential lower bound on the size of algebraic decision trees for MAX,” preprint, December 1995.Google Scholar
- [WY]A. Wigderson and A. Yao, manuscript, March 1996.Google Scholar