图形资源分配中的PROPm与PROPavg问题解决
Settling PROPm and PROPavg in Graphical Resource Allocation
- The Hong Kong Polytechnic University（香港理工大学）
- Shandong University（山东大学）
- University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决了图形资源分配中PROPm与PROPavg的开放问题：证明PROPm方向对多重图商品等总存在且可高效计算，而PROPavg可能不存在且判定为NP完全，并给出PROPm价格为2的常数界。
AI中文摘要：
我们研究了图形资源分配中的比例公平性，其中智能体是顶点，不可分割的物品是边，每个物品必须分配给其两个端点之一。已有研究证明PROP1方向总是存在，而PROPx方向可能不存在，但中间松弛PROPm和PROPavg（两者在没有图形约束的情况下均可满足）是否总能被满足的问题一直悬而未决。在本文中，我们解决了这一空白。我们证明了对于多重图上的商品，PROPm方向总是存在且可高效计算。该保证扩展到杂务以及商品与杂务的混合情形。与此形成鲜明对比的是，我们证明了即使对于具有二值估值的简单图，PROPavg方向也可能不存在，并且判定其存在性是NP完全的。我们进一步量化了PROPm的效率损失：对于商品，PROPm的价格恰好为2，这是少数能获得公平价格常数界的场景之一。我们通过PROPm下福利优化的困难性结果以及同时满足PROPm和帕累托最优的方向存在性的困难性结果，对这些结论进行了补充。
英文摘要：
We study proportional fairness in graphical resource allocation, where agents are vertices, indivisible items are edges, and each item must be allocated to one of its two endpoints. It has been proved that PROP1 orientations always exist and PROPx orientations may not, but it has remained open whether the intermediate relaxations PROPm and PROPavg (both can be satisfied without graphical constraints) can always be satisfied. In this paper, we resolve this gap. We prove that PROPm orientations always exist for goods on multigraphs and can be computed efficiently. The guarantee extends to chores and a mixture of goods and chores. In sharp contrast, we show that PROPavg orientations need not exist, even for simple graphs with binary valuations, and that deciding their existence is NP-complete. We further quantify the efficiency loss of PROPm: for goods, the price of PROPm is exactly $2$, which is one of the few settings where a constant bound on the price of fairness can be obtained. We complement these results with hardness results for welfare optimization under PROPm and for the existence of orientations that are simultaneously PROPm and Pareto optimal.
