Matroid Bayesian Online Selection

TL;DR

This work analyzes Bayesian online selection under matroid constraints, distinguishing between laminar and graphic matroids. It proves a Polynomial-Time Approximation Scheme (PTAS) for left-to-right laminar Bayesian selection while establishing PSPACE-hardness for graphic matroid Bayesian selection, showing that arrival orders cannot remove the hardness in the graphic case. The approach hinges on a LP-based relaxation with a big/small bin decomposition, concentration arguments under left-to-right arrivals, and a careful transformation analysis, with extensions to orders close to left-to-right. The results demonstrate that laminar structures admit tractable online approximations under natural arrival orders, whereas graphic structures remain intractable, underscoring a deep separation between matroid classes in online revenue optimization.

Abstract

We study a class of Bayesian online selection problems with matroid constraints. Consider a vendor who has several items to sell, with the set of sold items being subject to some structural constraints, e.g., the set of sold items should be independent with respect to some matroid. Each item has an offer value drawn independently from a known distribution. Given distribution information for each item, the vendor wishes to maximize their expected revenue by carefully choosing which offers to accept as they arrive. Such problems have been studied extensively when the vendor's revenue is compared with the offline optimum, referred to as the "prophet". In this setting, a tight 2-competitive algorithm is known when the vendor is limited to selling independent sets from a matroid [Kleinberg and Weinberg, 2012]. We turn our attention to the online optimum, or "philosopher", and ask how well the vendor can do with polynomial-time computation, compared to a vendor with unlimited computation but with the same limited distribution information about offers. We show that when the underlying constraints are laminar and the arrival of buyers follows a natural "left-to-right" order, there is a Polynomial-Time Approximation Scheme for maximizing the vendor's revenue. We also show that such a result is impossible for the related case when the underlying constraints correspond to a graphic matroid. In particular, it is $\texttt{PSPACE}$-hard to approximate the philosopher's expected revenue to some fixed constant $α< 1$; moreover, this cannot be alleviated by requirements on the arrival order in the case of graphic matroids.

Figures (6)

• Figure 1: Here, the horizontal axis is associated with clients and so with the timestamps, since their arrivals provide a measure for time. The vertical axis is associated with the value of the parameter $p$. The critical thresholds $p_1$, $p_2$, $p_3$ and $p_4$ for the value of $p$ are depicted on the vertical axis. Below the picture of the graph, one can find the illustration for the corresponding laminar matroid. In particular, the picture below contains an interval for each critical threshold and the timestamps, when the value of $p$ drops below the threshold and the next timestamp when the value of $p$ reaches the value of the threshold.
• Figure 2: Here the legend of the figure is the same as in Figure \ref{['fig:level_sets']}. The structure of the function for the parameter $p$ corresponds to the situation when $p$ represents a resource that is being delivered over time, and is spent only on servicing clients. In this scenario, the vendor can serve only a certain number of clients until the next delivery of the resource.
• Figure 3: Here, the vendor receives four clients $u_1$, $u_2$, $u_3$ and $u_4$ in the corresponding order, but can serve at most $2$ clients. The distributions for the offers are as in the figure, e.g., $u_2$ offers $3$ with probability $0.5$ and otherwise offers $0$, $u_4$ always offers $1$. Let $X_i$, $i=1,\ldots,4$ be the event (and the corresponding indicator variable) that $u_i$ was served by the optimal online policy. We have $\textbf{Pr}[X_3]=3/4$, $\textbf{Pr}[X_4]=1/4$ while $\textbf{Pr}[X_3\land X_4]=1/4$. Thus, we have $\textbf{Cov}(X_3,X_4)=1/16$.
• Figure 4: Here, the vendor receives six clients $u_1$,…, $u_6$ in the corresponding order, but can serve at most $3$ clients. Moreover, among $u_3$ and $u_4$ the vendor can serve at most one client. The distributions for the offers are as on the figure, e.g. $u_2$ makes offer $3$ with probability $0.5$ and otherwise makes offer $0$, $u_5$ always offers $1$. Let $X_i$, $i=1,\ldots,6$ be the event (and the corresponding indicator variable) that $u_i$ was served by the optimal online policy. We have $\textbf{Pr}[X_1]=\textbf{Pr}[X_2]=1/2$, $\textbf{Pr}[X_3]=1/4$, $\textbf{Pr}[X_4]=3/8$, $\textbf{Pr}[X_1\land X_4]=\textbf{Pr}[X_2\land X_4]=1/4$ and $\textbf{Pr}[X_3\land X_4]=0$. Thus, we have $\textbf{Cov}(X_1+X_2+X_3,X_4)=1/32$.
• Figure 5: The laminar family, capacities, and arrival order generated by Algorithm \ref{['alg:anticoncentrated']} when $r = 2$.

Theorems & Definitions (50)

• definition thmcounterdefinition
• lemma thmcounterlemma
• proof
• corollary thmcountercorollary
• lemma thmcounterlemma
• proof : Proof of Lemma \ref{['lem:logdepth']}
• corollary thmcountercorollary
• proof
• proposition thmcounterproposition: Proposition 2.1 from AnariLMBS
• proof