Planning Against a Prophet: A Graph-Theoretic Framework for Making Sequential Decisions

TL;DR

A general graph-theoretic framework that captures many well-studied prophet inequality problems, including d-dimensional matching, k-prophet inequality, and more, and generalizes it to accommodate correlations between edges originating from the same node and allow for additional constraints on the edges the agent can take.

Abstract

We devise a general graph-theoretic framework for studying prophet inequalities. In this framework, an agent traverses a directed acyclic graph from a starting node $s$ to a target node $t$. Each edge has a value that is sampled from a known distribution. When the agent reaches a node $v$ it observes the realized values of all the outgoing edges from $v$. The agent's objective is to maximize the expected total value of the path it takes. As in prophet inequalities, we compare the agent's performance against a prophet who observes all the realizations of the edges' values ahead of time. Our analysis reveals that this ratio highly depends on the number of paths $k$ required to cover all the nodes in the graph. In particular, we provide an algorithm that guarantees a prophet inequality ratio of $\frac{1}{2k}$ and show an upper bound of $\frac{1}{k+1}$. Our framework captures planning problems in which there is uncertainty regarding the costs/benefits of each action. In particular, it captures an over-time variant of the classic prophet inequality in which a seller leases a durable item, such as an apartment, for $n$ time units. Each period a lessee appears and may have a different value for each lease term. We obtain a tight bound of $1/2$ for this variant. To make this framework even more expressive, we further generalize it to accommodate correlations between edges originating from the same node and allow for additional constraints on the edges the agent can take. The generalized framework captures many well-studied prophet inequality problems, including $d$-dimensional matching, $k$-prophet inequality, and more.

Figures (7)

• Figure 1: Graph for capturing classic prophet inequality. The starting node is $s=1$.
• Figure 2: An illustration for the over-time variant of prophet inequality with a time horizon of $5$ steps. Some of the edges are gray to make the illustration easier to digest.
• Figure 3: An illustration for over-time prophet inequality with two markets. To simplify the illustration the candidates can be hired for either one time step or two. From $s$, the DM can decide which market to go to first so both edges have weight $0$. For any $i$ the weight of the edges $(v_i,v_{i+1}), (v_i,u_{i+1})$ is the same and the weight of the edges $(v_i,v_{i+2}), (v_i,u_{i+2})$ is the same.
• Figure 4: Instance from \ref{['ex:49k1d1']} that attains the upper bound of $4/9$ for graphs with a single focal path and $d=1$. $B_x$ denotes an independent realization of a variable that has value $1/x$ with probability $x$, and $0$ otherwise. There is only one label, red, and the thick red arrows represent edges with this label. The capacity of the red label is $1$.
• Figure 5: An illustration of the construction of a graph $G_i$ in the proof of \ref{['thm:general_graphs']}. We start with a path $P_i=(s,2,3,4,t)$ and replace edges that leave the path (thick grey arrows, $a,b,c$, and $d$) with an artificial edge (dashed arrows, $a',b',c'$, and $d'$) that ends in the earliest node in $P_i$ that can be reached after taking the original edge. The values of the artificial edges are the same as in the original edges, and we ignore the values of the intermediate edges that do not start from nodes in $P_i$.

Theorems & Definitions (18)

• Theorem 1: Dilworth's Theorem D50
• Theorem 2
• Lemma 1
• proof
• proof : Proof of \ref{['thm:width1_nolabel']}.
• Theorem 3
• Lemma 2
• proof
• proof : Proof of \ref{['thm:colors']}.
• Example 3.1: Upper bound of $4/9$ for $k=1$ and $d=1$