Tomasz Ponitka

Michal Feldman, Jugal Garg, Vishnu V. Narayan et al.

We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an $Ω(\log m/ \log \log m)$-approximation to the optimal makespan, where $m$ is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a $3/2$-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.

Paul Duetting, Michal Feldman, Tomasz Ponitka et al.

Algorithmic contract design studies scenarios where a principal incentivizes an agent to exert effort on her behalf. In this work, we focus on settings where the agent's type is drawn from an unknown distribution, and formalize an offline learning framework for learning near-optimal contracts from sample agent types. A central tool in our analysis is the notion of pseudo-dimension from statistical learning theory. Beyond its role in establishing upper bounds on the sample complexity, pseudo-dimension measures the intrinsic complexity of a class of contracts, offering a new perspective on the tradeoffs between simplicity and optimality in contract design. Our main results provide essentially optimal tradeoffs between pseudo-dimension and representation error (defined as the loss in principal's utility) with respect to linear and bounded contracts. Using these tradeoffs, we derive sample- and time-efficient learning algorithms, and demonstrate their near-optimality by providing almost matching lower bounds on the sample complexity. Conversely, for unbounded contracts, we prove an impossibility result showing that no learning algorithm exists. Finally, we extend our techniques in three important ways. First, we provide refined pseudo-dimension and sample complexity guarantees for the combinatorial actions model, revealing a novel connection between the number of critical values and sample complexity. Second, we extend our results to menus of contracts, showing that their pseudo-dimension scales linearly with the menu size. Third, we adapt our algorithms to the online learning setting, where we show that, a polynomial number of type samples suffice to learn near-optimal bounded contracts. Combined with prior work, this establishes a formal separation between expert advice and bandit feedback for this setting.

Jarosław Byrka, Franciszek Malinka, Tomasz Ponitka

We study the fair division of indivisible items and provide new insights into the EFX problem, which is widely regarded as the central open question in fair division, and the PMMS problem, a strictly stronger variant of EFX. Our first result constructs a three-agent instance with two monotone valuations and one additive valuation in which no PMMS allocation exists. Since EFX allocations are known to exist under these assumptions, this establishes a formal separation between EFX and PMMS. We prove existence of fair allocations for three important special cases. We show that EFX allocations exist for personalized bivalued valuations, where for each agent $i$ there exist values $a_i > b_i$ such that agent $i$ assigns value $v_i(\{g\}) \in \{a_i, b_i\}$ to each good $g$. We establish an analogous existence result for PMMS allocations when $a_i$ is divisible by $b_i$. We also prove that PMMS allocations exist for binary-valued MMS-feasible valuations, where each bundle $S$ has value $v_i(S) \in \{0, 1\}$. Notably, this result holds even without assuming monotonicity of valuations and thus applies to the fair division of chores and mixed manna. Finally, we study a class of valuations called pair-demand valuations, which extend the well-studied unit-demand valuations to the case where each agent derives value from at most two items, and we show that PMMS allocations exist in this setting. Our proofs are constructive, and we provide polynomial-time algorithms for all three existence results.