On a search problem related to branch-and-bound procedures

o. INTRODUCTION Branch-and-bound procedures are commonly used in practice for the solution of NP-hard combinatorial optimization problems (see e.g., [PS, LW]). We describe a model that captures the essential elements common to all branch-and-bound procedures. Storage limitations in these procedures raise in a natural wayan interesting search problem. We present deterministic and probabilistic algorithms for this problem. These imply that branch-and-bound can be performed with very small storage and only slightly superlinear time. In section 1 we describe the search problem and state our results. In section 2 we describe the algorithms. Section 3 elaborates on the relation of our search problem to branch-and-bound procedures. The input will be a valued, ordered, infinite binary tree T. By this we mean (i) every vertex vET has a left child and a right child, denoted L(v) and R(u) respectively. The root of T is r. every vertex has a value, val(v), and the function val: T-+ N satisfies val(v) S val(L(v» and val(v) S val(R(u). (In other words, values along any path from the root increase, so the tree is a heap). (For simplicity we shall assume that values on the tree are distinct, i.e.