resource allocation planners

Abstract: Given $n$ points in a metric space, partitioned into groups, $X_1,\dots,X_m$, and integer quotas, $k_1,\dots,k_m$, summing to $k$, the Fair Max-Min Diversification problem asks for a set of $k$ points, exactly $k_i$ from each group $X_i$, maximizing the minimum pairwise distance. Addanki et al. (ICDT 2022) described a ball LP for this problem and rounding algorithms that yield a factor 2 approximation whose fairness holds only in expectation and a factor 6 approximation with relaxed fairness guarantees, as well as an $(m+1)$-approximation with exact fairness. We introduce two new algorithms. The first method refines the rounding of Addanki et al., yielding a 2-approximate solution that is $\varepsilon$-fair with high probability, meaning that from every group $X_i$, at least $(1-\varepsilon) k_i$ points are chosen. The second method in polynomial time returns a 4-approximation to the optimal value of the exact fairness version. In time $n^{O(1)} 2^{O(k)}$, which is fixed-parameter tractable in $k$, we achieve an exactly fair 4-approximation. Our method adapts the augmenting procedure behind Haxell's theorem (Graphs Combin., 1995). This approximation factor does not depend on $m$. Moreover, we show that no rounding of the ball LP achieves a smaller factor with exact fairness.

Abstract: In this paper, we prove the existence of an envy-free up to one item (EF1) division for a discrete chore. Our approach builds on the powerful framework of Simmons-Su, which leverages Sperner's lemma to guarantee the existence of a simplex corresponding to a sequence of similar fractional divisions, ensuring that each agent is satisfied with a different bundle. Bilò et al. [2022] introduced a rounding technique that converts the fractional divisions into a connected integral EF1 division for goods when there are at most four agents, and this method was later extended by Igarashi [2023] to accommodate any number of agents. However, these rounding techniques for goods do not directly apply to chores because the definitions of EF1 differ in the two settings. To overcome this asymmetry, we modify the existing rounding techniques and show that connected EF1 divisions exist for a discrete chore.