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• W4376504100 abstract "Abstract This 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 $$smash {2^{frac{1}{5}n}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mpadded> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mpadded> </mml:math> in time $$O(2^{delta n})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$delta <tfrac{1}{5}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo><</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:mstyle> </mml:mrow> </mml:math> , unless the strong exponential time hypothesis fails. Similarly, for any $$varepsilon > 0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , no polynomial time $$smash {2^{(frac{1}{2} - varepsilon )n}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mpadded> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mi>ε</mml:mi> <mml:mo>)</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mpadded> </mml:math> -approximation is possible, unless $$text {P} = text {NP} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtext>P</mml:mtext> <mml:mo>=</mml:mo> <mml:mtext>NP</mml:mtext> </mml:mrow> </mml:math> . We also show that computing the size of a smallest branch-and-bound tree exactly is $${#P} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>#</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> -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." @default.
• W4376504100 created "2023-05-15" @default.
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• W4376504100 date "2023-05-13" @default.
• W4376504100 modified "2023-10-14" @default.
• W4376504100 title "On computing small variable disjunction branch-and-bound trees" @default.
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• W4376504100 doi "https://doi.org/10.1007/s10107-023-01968-y" @default.
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