We introduce a novel collective decision making problem that captures the ubiquitous issue of ordering food to cater for varied dietary preferences and requirements. Our settings involve agents with diverse dietary requirements over menu options with varied serving sizes. The goal is to select a menu where everyone has enough food they can consume and wastage of food is minimized. We introduce two different consumption models: optimistic and pessimistic. Optimistic consumption assumes a situation when a central planner can optimally allocate the food ordered among the agents to maximize the number of people who get enough to eat. Pessimistic considers the worst case guarantee on consumption when agents fill their own plates in an arbitrary order. Under either consumption model, we seek valid menus (under which all agents are sufficiently fed) of minimum size. Our work provides two sets of characterizations: (1) we characterize valid menus under either consumption model and (2) we characterize the space of instances that admit polynomial-time algorithms to find minimum sized menus. Our results also help us design Integer Linear Programs to find minimum sized menus in general settings. Furthermore, we present polynomial-time algorithms for important special cases. We then consider the worst case discrepancy between the size of minimum sized optimistic and pessimistic menus. We call this the waste of pessimism, captured by the ratio of the minimum sized pessimistic menu to that of the minimum sized optimistic menu. We show tight upper bounds on this ratio. Our results also provide additional insights on the problem of finding a minimum sized maximal matching, which may be of independent interest.

翻译：我们提出了一种新颖的集体决策问题，该问题捕捉了为满足多样化饮食偏好和需求而订购食物的普遍困境。我们的设置涉及具有不同饮食需求的个体，以及提供多种份量选项的菜单。目标是选择一份菜单，使得每个人都有足够的可食用食物，同时最大限度地减少浪费。我们引入了两种不同的消费模型：乐观模型和悲观模型。乐观消费假设存在一个中央规划者能够最优地在个体间分配所订食物，以最大化获得足够食物的人数。悲观消费则考虑了个体以任意顺序自行取餐时，关于食物消耗的最坏情况保证。在这两种消费模型下，我们寻求有效菜单（即所有个体都能充分饱食的菜单）的最小规模。我们的工作提供了两组特征刻画：(1) 刻画了两种消费模型下的有效菜单；(2) 刻画了能够通过多项式时间算法找到最小规模菜单的实例空间。我们的结果还帮助我们设计出在一般情况下寻找最小规模菜单的整数线性规划。此外，我们为重要的特殊情形提出了多项式时间算法。接着，我们考虑了乐观与悲观菜单最小规模之间的最坏情况差异。我们将此称为“悲观代价”，并以悲观菜单最小规模与乐观菜单最小规模的比值来衡量。我们证明了该比值的紧上界。我们的结果还为寻找最小规模极大匹配问题提供了额外见解，这可能具有独立的研究价值。