The Keychain Problem: On Minimizing the Opportunity Cost of Uncertainty
Ramiro N. Deo-Campo Vuong, Robert Kleinberg, Aditya Prasad, Eric Xiao, Haifeng Xu
TL;DR
The paper introduces the Keychain Problem, a sequential decision problem with limited action availability and Bayesian priors, and studies Bayes-optimal policies across variants. It establishes a core algorithmic bridge by reducing the basic problem to Maximum Weight Bipartite Matching ($MWBM$) and the probabilistic-order variant to Maximum Weight Laminar Matching ($MWLM$), which further reduces to Combinatorial Auctions with XOS valuations, enabling a $(1-1/e)$-approximation in these settings. It also proves hardness results for probabilistic scenarios (APX-hard) and for the optimizable-order variant (NP-hardness) while offering a simple $(1/2)$-approximation for order selection. The work further connects these findings to online bipartite matching and the Philosopher Inequality, and discusses black-box priors, with applications to clinical trials and camouflaged cargo transport, providing a structural framework linking sequential decision-making under limited availability to classical graph- and auction-theoretic problems. Overall, the paper advances both the theory and potential practice of efficient exploration under constrained availability through principled reductions and approximation results.
Abstract
In this paper, we introduce a family of sequential decision-making problems, collectively termed the Keychain Problem, that involve exploring a set of actions to maximize expected payoff when only a subset of actions are available in each stage. In an instance of the Keychain Problem, a locksmith faces a sequence of decisions, each of which involves selecting one key from a keychain (a subset of keys) to attempt to open a lock. Given a Bayesian prior on the effectiveness of keys, the locksmith's goal is to minimize the opportunity cost, which is the expected number of rounds in which the chain has a correct key but our selected key is incorrect. We study the computation of the Bayes optimal solution for Keychain Problems. Employing polynomial-time reductions, we establish formal connections between natural variants of the Keychain Problem and well-studied algorithmic economics problems on bipartite graphs. When the keychain order is known to the locksmith, we show that it reduces to Maximum Weight Bipartite Matching (MWBM). More general is the situation when the keychain order is sampled from a prior distribution (possibly correlated with the correct key). Here the Keychain Problem reduces to a novel generalization of MWBM which we coin the Maximum Weight Laminar Matching, which then further reduces to combinatorial auctions under XOS valuation functions. Finally, we show that when the locksmith can choose the keychain order, the Keychain problem reduces from a classic NP-hard combinatorial problem, again, on bipartite graphs. Besides implying algorithmic results and deepening our structural understanding about the Keychain Problem, our established reductions also find applications beyond -- for example, to the Philosopher Inequality for online bipartite matching.