We study the problem of allocating indivisible items on a path among agents. The objective is to find a fair and efficient allocation in which each agent's bundle forms a contiguous block on the line. We demonstrate that, even when the valuations are binary additive, deciding whether every item can be allocated to an agent who wants it is NP-complete. Consequently, we provide two fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, with respect to the number of agents and the number of items. Additionally, we present a 2-approximation algorithm for the special case when the valuations are binary additive and the maximum utility is equal to the number of items. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least 2 or at most 1 is NP-complete, even when the valuations are binary additive. We also explore the case where the order of the blocks of items allocated to the agents is predetermined. In this case, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. However, we determine that checking the existence of an EF1 allocation is NP-complete, even when the valuations are binary additive.

翻译：我们研究在一条路径上向参与者分配不可分物品的问题。目标是找到一种公平且高效的分配方案，使得每个参与者获得的物品集合在线上形成一个连续区块。我们证明，即使估值是二进制可加的，判定每个物品是否能分配给想要它的参与者的问题是NP完全的。因此，我们提供了两种关于参与者数量和物品数量的固定参数可解（FPT）算法，用于最大化功利主义社会福利。此外，针对估值是二进制可加且最大效用等于物品数量的特殊情形，我们提出了一种2-近似算法。进一步，我们证明即使估值是二进制可加的，判定最大平均主义社会福利是否至少为2或至多为1的问题是NP完全的。我们还探讨了分配给参与者的物品区块顺序预定的情形。在这种情况下，我们表明最大功利主义社会福利和平均主义社会福利均可在多项式时间内计算。然而，我们判定，即使估值是二进制可加的，检查EF1分配的存在性仍然是NP完全的。