[Paper Review] Think Eternally: Improved Algorithms for the Temp Secretary Problem and Extensions

Thomas Keßelheim, Andreas Tönnis|arXiv (Cornell University)|Jun 22, 2016

Optimization and Search Problems11 references3 citations

TL;DR

This paper introduces a novel relaxation-based algorithm for the Temp Secretary Problem with temporary commitments, transforming temporal constraints into global ones to enable efficient online decision-making. It achieves a competitive ratio of $\frac{1}{2} - O(\sqrt{\gamma})$ for general cases and $1 - O\left(\frac{1}{\sqrt{B}}\right) - O(\sqrt{\gamma})$ for large $B$, significantly improving prior bounds while extending to general packing constraints and variable-duration contracts.

ABSTRACT

The \emph{Temp Secretary Problem} was recently introduced by Fiat et al. It is a generalization of the Secretary Problem, in which commitments are temporary for a fixed duration. We present a simple online algorithm with improved performance guarantees for cases already considered by Fiat et al.\ and give competitive ratios for new generalizations of the problem. In the classical setting, where candidates have identical contract durations $γ\ll 1$ and we are allowed to hire up to $B$ candidates simultaneously, our algorithm is $(\frac{1}{2} - O(\sqrtγ))$-competitive. For large $B$, the bound improves to $1 - O\left(\frac{1}{\sqrt{B}} ight) - O(\sqrtγ)$. Furthermore we generalize the problem from cardinality constraints towards general packing constraints. We achieve a competitive ratio of $1 - O\left(\sqrt{\frac{(1+\log d + \log B)}{B}} ight) -O(\sqrtγ)$, where $d$ is the sparsity of the constraint matrix and $B$ is generalized to the capacity ratio of linear constraints. Additionally we extend the problem towards arbitrary hiring durations. Our algorithmic approach is a relaxation that aggregates all temporal constraints into a non-temporal constraint. Then we apply a linear scaling algorithm that, on every arrival, computes a tentative solution on the input that is known up to this point. This tentative solution uses the non-temporal, relaxed constraints scaled down linearly by the amount of time that has already passed.

Motivation & Objective

• To address the Temp Secretary Problem where hiring commitments last for a fixed duration $\gamma \ll 1$, and at most $B$ items can overlap at any time.
• To improve competitive ratios beyond prior work, especially for large $B$ and small $\gamma$, in the stochastic arrival model with uniform random arrival times.
• To generalize the cardinality constraint to arbitrary linear packing constraints, enabling applications like multiple knapsack or machine scheduling.
• To extend the model to items with arbitrary durations, not just fixed $\gamma$, and analyze the performance under such generalizations.
• To provide a unified algorithmic framework that relaxes temporal constraints into global ones, enabling simpler and more robust online decision-making.

Proposed method

• Propose a relaxation of the original problem by replacing temporal overlap constraints with a global cap of $B\lceil 1/\gamma \rceil$ items over the entire horizon.
• Use a linear scaling rule: at time $t$, scale down the relaxed constraints by the fraction of time elapsed, $t$, to maintain feasibility.
• Apply an online algorithm for linear packing problems to the relaxed, time-scaled constraints to compute a tentative solution at each arrival.
• Derive the final selection by accepting items only if they are feasible in the tentative solution and do not violate the original temporal constraints.
• Analyze the competitive ratio via probabilistic bounds on feasibility and selection, using concentration inequalities and conditional expectations.
• Generalize the approach to arbitrary item durations by adapting the relaxation and scaling mechanism to variable-length contracts.

Experimental results

Research questions

• RQ1 Can we design a simpler and more effective online algorithm for the Temp Secretary Problem with temporary commitments that improves upon the best-known competitive ratio?
• RQ2 What competitive ratio can be achieved when $B$ is large, and how does it scale with $\gamma$ and $B$?
• RQ3 Can the cardinality constraint in the Temp Secretary Problem be generalized to arbitrary linear packing constraints without sacrificing competitiveness?
• RQ4 How does the competitive ratio behave when items have variable durations rather than fixed $\gamma$?
• RQ5 Can the relaxation-based approach be extended to other feasibility structures, such as matroids or submodular objectives?

Key findings

• The proposed algorithm achieves a competitive ratio of $\frac{1}{2} - O(\sqrt{\gamma})$ for all $B$, improving upon the prior bound of $\frac{1}{2} - O\left(\sqrt{\gamma \ln(1/\gamma)}\right)$ for $B=1$.
• For large $B$, the competitive ratio improves to $1 - O\left(\frac{1}{\sqrt{B}}\right) - O(\sqrt{\gamma})$, outperforming the previous $1 - O\left(\sqrt{\frac{\ln B}{B}}\right) - O\left(\sqrt{\gamma \ln(1/\gamma)}\right)$ result.
• The algorithm generalizes to arbitrary linear packing constraints with a competitive ratio of $1 - O\left(\sqrt{\frac{1 + \log d + \log B}{B}}\right) - O(\sqrt{\gamma})$, where $d$ is the sparsity of the constraint matrix.
• The method extends naturally to items with arbitrary durations, achieving a competitive ratio of $\frac{1}{4} - \Theta(\sqrt{\gamma})$ in the general case.
• The relaxation technique effectively decouples temporal constraints from the online decision process, enabling tighter analysis and better performance guarantees.
• The analysis relies on bounding the probability of infeasibility in each round using conditional expectations and concentration bounds, leading to a clean and scalable competitive ratio derivation.