# Hypergraphs and decision trees

- First Online:

DOI: 10.1007/3-540-62559-3_1

- Cite this paper as:
- Yao A.C. (1997) Hypergraphs and decision trees. In: d'Amore F., Franciosa P.G., Marchetti-Spaccamela A. (eds) Graph-Theoretic Concepts in Computer Science. WG 1996. Lecture Notes in Computer Science, vol 1197. Springer, Berlin, Heidelberg

## 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.