Online Envy Minimization and Multicolor Discrepancy: Equivalences and Separations

TL;DR

This paper establishes the precise relationship between online envy minimization and multicolor discrepancy, proving tight bounds and separations under different adversarial models, thereby resolving key open problems in the field.

Contribution

It proves the equivalence of envy minimization and multicolor discrepancy against an oblivious adversary and demonstrates a separation under an i.i.d. adversary, providing tight bounds and resolving open questions.

Findings

Established a $O( ootlog T)$ upper bound for multicolor discrepancy.

Proved a $ ext{Omega}( ootlog T)$ lower bound for envy minimization.

Identified a separation with a lower bound for vector balancing and a constant upper bound for envy minimization under i.i.d. adversaries.

Abstract

We consider the fundamental problem of allocating $T$ indivisible items that arrive over time to $n$ agents with additive preferences, with the goal of minimizing envy. This problem is tightly connected to online multicolor discrepancy: vectors $v_1, \dots, v_T \in \mathbb{R}^d$ with $\| v_i \|_2 \leq 1$ arrive over time and must be, immediately and irrevocably, assigned to one of $n$ colors to minimize $\max_{i,j \in [n]} \| \sum_{v \in S_i} v - \sum_{v \in S_j} v \|_{\infty}$ at each step, where $S_\ell$ is the set of vectors that are assigned color $\ell$. The special case of $n = 2$ is called online vector balancing. Any bound for multicolor discrepancy implies the same bound for envy minimization. Against an adaptive adversary, both problems have the same optimal bound, $\Theta(\sqrt{T})$, but whether this holds for weaker adversaries is unknown. Against an oblivious adversary, Alweiss et al. give a $O(\log T)$ bound, with high probability, for multicolor discrepancy. Kulkarni et al. improve this to $O(\sqrt{\log T})$ for vector balancing and give a matching lower bound. Whether a $O(\sqrt{\log T})$ bound holds for multicolor discrepancy remains open. These results imply the best-known upper bounds for envy minimization (for an oblivious adversary) for $n$ and two agents, respectively; whether better bounds exist is open. In this paper, we resolve all aforementioned open problems. We prove that online envy minimization and multicolor discrepancy are equivalent against an oblivious adversary: we give a $O(\sqrt{\log T})$ upper bound for multicolor discrepancy, and a $\Omega(\sqrt{\log T})$ lower bound for envy minimization. For a weaker, i.i.d. adversary, we prove a separation: For online vector balancing, we give a $\Omega\left(\sqrt{\frac{\log T}{\log \log T}}\right)$ lower bound, while for envy minimization, we give an algorithm that guarantees a constant upper bound.