Graph Pricing with Limited Supply

We study approximation algorithms for graph pricing with vertex capacities yet without the traditional envy-free constraint. Specifically, we have a set of items $V$ and a set of customers $X$ where each customer $i \in X$ has a budget $b_i$ and is interested in a bundle of items $S_i \subseteq V$ with $|S_i| \leq 2$. However, there is a limited supply of each item: we only have $\mu_v$ copies of item $v$ to sell for each $v \in V$. We should assign prices $p(v)$ to each $v \in V$ and chose a subset $Y \subseteq X$ of customers so that each $i \in Y$ can afford their bundle ($p(S_i) \leq b_i$) and at most $\mu_v$ chosen customers have item $v$ in their bundle for each item $v \in V$. Each customer $i \in Y$ pays $p(S_i)$ for the bundle they purchased: our goal is to do this in a way that maximizes revenue. Such pricing problems have been studied from the perspective of envy-freeness where we also must ensure that $p(S_i) \geq b_i$ for each $i \notin Y$. However, the version where we simply allocate items to customers after setting prices and do not worry about the envy-free condition has received less attention. Our main result is an 8-approximation for the capacitated case via local search and a 7.8096-approximation in simple graphs with uniform vertex capacities. The latter is obtained by combing a more involved analysis of a multi-swap local search algorithm for constant capacities and an LP-rounding algorithm for larger capacities. If all capacities are bounded by a constant $C$, we further show a multi-swap local search algorithm yields an $\left(4 \cdot \frac{2C-1}{C} + \epsilon\right)$-approximation. We also give a $(4+\epsilon)$-approximation in simple graphs through LP rounding when all capacities are very large as a function of $\epsilon$.