A new lower bound for multi-color discrepancy with applications to fair division
Ioannis Caragiannis, Kasper Green Larsen, Sudarshan Shyam
TL;DR
This work advances discrepancy theory and fair division by establishing a new lower bound for multi-color discrepancy: DISC(n,k) = Ω(√(n/ln k)). Using a probabilistic construction and reverse Chernoff bounds, the authors also derive improved infeasibility results for consensus 1/k-division up to d items (CD$d$), envy-freeness up to d items (EF$d$), and proportionality up to d items (PROP$d$), with bounds Ω(√(n/ln k)), Ω(√(n/(k ln k))), and Ω(√(n/(k^3 ln k))) respectively, including a min-group-size variant for PROP$d$. The methods avoid black-box reductions and extend to binary valuations, contributing tighter lower bounds than prior work and suggesting broader implications for related combinatorial and algorithmic problems. The results close some gaps but leave others open, notably the discrepancy gap by a factor of roughly √(ln k), and motivate further exploration of applications and tighter bounds.
Abstract
A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as possible. We present a new lower bound for multi-color discrepancy, showing that there is a set system with $n$ subsets over a set of elements in which any $k$-coloring of the elements has discrepancy at least $Ω\left(\sqrt{\frac{n}{\ln{k}}}\right)$. This result improves the previously best-known lower bound of $Ω\left(\sqrt{\frac{n}{k}}\right)$ of Doerr and Srivastav [2003] and may have several applications. Here, we explore its implications on the feasibility of fair division concepts for instances with $n$ agents having valuations for a set of indivisible items. The first such concept is known as consensus $1/k$-division up to $d$ items (\cd$d$) and aims to allocate the items into $k$ bundles so that no matter which bundle each agent is assigned to, the allocation is envy-free up to $d$ items. The above lower bound implies that \cd$d$ can be infeasible for $d\in Ω\left(\sqrt{\frac{n}{\ln{k}}}\right)$. We furthermore extend our proof technique to show that there exist instances of the problem of allocating indivisible items to $k$ groups of $n$ agents in total so that envy-freeness and proportionality up to $d$ items are infeasible for $d\in Ω\left(\sqrt{\frac{n}{k\ln{k}}}\right)$ and $d\in Ω\left(\sqrt{\frac{n}{k^3\ln{k}}}\right)$, respectively. The lower bounds for fair division improve the currently best-known ones by Manurangsi and Suksompong [2022].