On computing small variable disjunction branch-and-bound trees

Abstract

AbstractThis article investigates smallest branch-and-bound trees and their computation. We first revisit the notion of hiding sets to deduce lower bounds on the size of branch-and-bound trees for certain binary programs, using both variable disjunctions and general disjunctions. We then provide exponential lower bounds for variable disjunctions by a disjoint composition of smaller binary programs. Moreover, we investigate the complexity of finding small branch-and-bound trees using variable disjunctions: We show that it is not possible to approximate the size of a smallest branch-and-bound tree within a factor of $$\smash {2^{\frac{1}{5}n}}$$ 2 1 5 n in time $$O(2^{\delta n})$$ O ( 2 δ n ) with $$\delta <\tfrac{1}{5}$$ δ < 1 5 , unless the strong exponential time hypothesis fails. Similarly, for any $$\varepsilon > 0$$ ε > 0 , no polynomial time $$\smash {2^{(\frac{1}{2} - \varepsilon )n}}$$ 2 ( 1 2 - ε ) n -approximation is possible, unless $$\text {P} = \text {NP} $$ P = NP . We also show that computing the size of a smallest branch-and-bound tree exactly is $${\#P} $$ # P -hard. Similar results hold for estimating the size of the tree produced by branching rules like most-infeasible branching. Finally, we discuss that finding small branch-and-bound trees generalizes finding short treelike resolution refutations, and thus non-automatizability results transfer from this setting.