arXiv subjects

Marco Schmalhofer

Publications and source records attributed to Marco Schmalhofer.

3 recordsLinked to original sources

arXiv2026

Maximizing Nash Social Welfare in 2-Value Instances: Delineating Tractability

We study the problem of allocating a set of indivisible goods among a set of agents with \emph{2-value additive valuations}. In this setting, each good is valued either $1$ or $p/q$, for some fixed co-prime numbers $p,q\in \mathbb{N}$ such that $1\leq q < p$. Our goal is to find an allocation maximizing the \emph{Nash social welfare} (\NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of \NSW\ maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum \NSW\ allocation when the valuation functions are \emph{integral}, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum \NSW\ allocation for the \emph{half-integral} case, that is, $q=2$. Finally, we show it is \classNP-hard to compute an allocation with maximum \NSW\ whenever $q\geq3$.

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arXiv2022

Best of Both Worlds: Agents with Entitlements

Fair division of indivisible goods is a central challenge in artificial intelligence. For many prominent fairness criteria including envy-freeness (EF) or proportionality (PROP), no allocations satisfying these criteria might exist. Two popular remedies to this problem are randomization or relaxation of fairness concepts. A timely research direction is to combine the advantages of both, commonly referred to as Best of Both Worlds (BoBW). We consider fair division with entitlements, which allows to adjust notions of fairness to heterogeneous priorities among agents. This is an important generalization to standard fair division models and is not well-understood in terms of BoBW results. Our main result is a lottery for additive valuations and different entitlements that is ex-ante weighted envy-free (WEF), as well as ex-post weighted proportional up to one good (WPROP1) and weighted transfer envy-free up to one good (WEF(1,1)). It can be computed in strongly polynomial time. We show that this result is tight - ex-ante WEF is incompatible with any stronger ex-post WEF relaxation. In addition, we extend BoBW results on group fairness to entitlements and explore generalizations of our results to instances with more expressive valuation functions.

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arXiv2021

Maximizing Nash Social Welfare in 2-Value Instances

We consider the problem of maximizing the Nash social welfare when allocating a set $\mathcal{G}$ of indivisible goods to a set $\mathcal{N}$ of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent $i \in \mathcal{N}$ for every good $j \in \mathcal{G}$ is $v_{ij} \in \{p,q\}$, for $p,q \in \mathbb{N}$, $p \le q$. Maybe surprisingly, we design an algorithm to compute an optimal allocation in polynomial time if $p$ divides $q$, i.e., when $p=1$ and $q \in \mathbb{N}$ after appropriate scaling. The problem is \classNP-hard whenever $p$ and $q$ are coprime and $p \ge 3$. In terms of approximation, we present positive and negative results for general $p$ and $q$. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is \classAPX-hard, with a lower bound of $1.000015$ achieved at $p/q = 4/5$.

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