The Knapsack Secretary Problem is Strictly Harder Than the Secretary Problem

2026-07-24

AI Summary

This work investigates whether the knapsack secretary problem can achieve the optimal competitive ratio of $1/e$ known for the classical secretary problem. By constructing hard instances and employing a combination of instance reduction, approximate sequential modeling, and linear programming duality analysis, the paper establishes—for the first time—a rigorous impossibility result: no algorithm can be $(1/e - 0.0001)$-competitive, thereby proving that the optimal competitive ratio for this problem is strictly less than $1/e$. Building on this theoretical limitation, the authors propose a novel algorithm that improves the best-known competitive ratio from 0.153 to 0.178, surpassing the previous performance barrier.

Abstract

The knapsack secretary problem is a generalization of the classical secretary problem where the accepted items must satisfy a knapsack constraint. A line of work has developed constant-competitive algorithms for this problem, with successive improvements culminating in the current best-known competitive ratio of $0.153$. A natural open question was whether the optimal $1/e$ competitive ratio for the classical secretary problem is also achievable for the knapsack secretary problem. We answer this question negatively by showing that no $(1/e - 0.0001)$-competitive algorithm exists for the knapsack secretary problem. The analysis of the family of hard instances we construct proceeds in three steps. First, we reduce the cardinal problem on these instances to an almost-ordinal problem. Second, we formulate a linear program that captures the performance of almost-ordinal algorithms on this instance family. Finally, we exhibit a feasible dual solution whose objective value is strictly below $1/e$. We also give an algorithm that improves the best-known competitive ratio from $0.153$ to $0.178$.

Problem

• Knapsack Secretary Problem
• Secretary Problem
• Competitive Ratio
• Online Algorithms
• Hardness

Innovation

• knapsack secretary problem
• competitive ratio
• linear programming duality
• online algorithms
• hardness result