Equitable Division of a Path

We study fair resource allocation under a connectedness constraint wherein a set of indivisible items are arranged on a \emph{path} and only connected subsets of items may be allocated to the agents. An allocation is deemed fair if it satisfies \emph{equitability up to one good} (EQ1), which requires that agents' utilities are approximately equal. We show that achieving EQ1 in conjunction with well-studied measures of \emph{economic efficiency} (such as Pareto optimality, non-wastefulness, maximum egalitarian or utilitarian welfare) is computationally hard even for \emph{binary} additive valuations. On the algorithmic side, we show that by relaxing the efficiency requirement, a connected EQ1 allocation can be computed in polynomial time for \emph{any} given ordering of agents, even for general monotone valuations. Interestingly, the allocation computed by our algorithm has the highest egalitarian welfare among all allocations consistent with the given ordering. On the other hand, if efficiency is required, then tractability can still be achieved for binary additive valuations with \emph{interval structure}. On our way, we strengthen some of the existing results in the literature for other fairness notions such as envy-freeness up to one good (EF1), and also provide novel results for negatively-valued items or \emph{chores}.