Contiguous allocation of indivisible items on a path

Abstract

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 examine two natural settings: the fixed-order setting, where the order is predetermined, and the flexible-order setting, where the sequence of agents is unrestricted. In the fixed-order setting, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. Moreover, allocations satisfying proportionality, equitability, and the maximin share guarantee can also be computed in polynomial time in this setting. In contrast, checking the existence of an EF1 allocation is NP-complete for binary additive valuations. In the flexible-order setting, we demonstrate that, even when the valuations are binary additive, deciding whether every item can be allocated to an agent who values it is NP-complete. Consequently, we provide two fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, parameterized by the number of agents and the number of items. Additionally, we present a \(\varvec{1/2}\)-approximation algorithm for a special case. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least \(\varvec{2}\) or at most \(\varvec{1}\) is NP-complete for binary additive valuations. We also give two FPT algorithms for maximizing egalitarian social welfare, parameterized by the number of agents and the number of items.