Chore division on a graph

The paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent's share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Chore division on a graph

Le PDF est une version non publiée datant de 2018.International audienceThe paper considers fair allocation of indivisible nondisposable items that generate disutility (chores). We assume that these items are placed in the vertices of a graph and each agent’s share has to form a connected subgraph of this graph. Although a similar model has been investigated before for goods, we show that the goods and chores settings are inherently different. In particular, it is impossible to derive the solution of the chores instance from the solution of its naturally associated fair division instance. We consider three common fair division solution concepts, namely proportionality, envy-freeness and equitability, and two individual disutility aggregation functions: additive and maximum based. We show that deciding the existence of a fair allocation is hard even if the underlying graph is a path or a star. We also present some efficiently solvable special cases for these graph topologies

Fair Allocation based on Diminishing Differences

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice, course allocations and residency matches. In some cases, several `items' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities. We also consider chore allocation under the analogous condition --- increasing-differences.Comment: Revised version, based on very helpful suggestions of JAIR referees. Gaps in some proofs were filled, more experiments were done, and mor

Algorithms for Competitive Division of Chores

We study the problem of allocating divisible bads (chores) among multiple agents with additive utilities, when money transfers are not allowed. The competitive rule is known to be the best mechanism for goods with additive utilities and was recently extended to chores by Bogomolnaia et al (2017). For both goods and chores, the rule produces Pareto optimal and envy-free allocations. In the case of goods, the outcome of the competitive rule can be easily computed. Competitive allocations solve the Eisenberg-Gale convex program; hence the outcome is unique and can be approximately found by standard gradient methods. An exact algorithm that runs in polynomial time in the number of agents and goods was given by Orlin. In the case of chores, the competitive rule does not solve any convex optimization problem; instead, competitive allocations correspond to local minima, local maxima, and saddle points of the Nash Social Welfare on the Pareto frontier of the set of feasible utilities. The rule becomes multivalued and none of the standard methods can be applied to compute its outcome. In this paper, we show that all the outcomes of the competitive rule for chores can be computed in strongly polynomial time if either the number of agents or the number of chores is fixed. The approach is based on a combination of three ideas: all consumption graphs of Pareto optimal allocations can be listed in polynomial time; for a given consumption graph, a candidate for a competitive allocation can be constructed via explicit formula; and a given allocation can be checked for being competitive using a maximum flow computation as in Devanur et al (2002). Our algorithm immediately gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy (2018).Comment: 38 pages, 4 figure

On the Existence of Competitive Equilibrium with Chores

We study the chore division problem in the classic Arrow-Debreu exchange setting, where a set of agents want to divide their divisible chores (bads) to minimize their disutilities (costs). We assume that agents have linear disutility functions. Like the setting with goods, a division based on competitive equilibrium is regarded as one of the best mechanisms for bads. Equilibrium existence for goods has been extensively studied, resulting in a simple, polynomial-time verifiable, necessary and sufficient condition. However, dividing bads has not received a similar extensive study even though it is as relevant as dividing goods in day-to-day life. In this paper, we show that the problem of checking whether an equilibrium exists in chore division is NP-complete, which is in sharp contrast to the case of goods. Further, we derive a simple, polynomial-time verifiable, sufficient condition for existence. Our fixed-point formulation to show existence makes novel use of both Kakutani and Brouwer fixed-point theorems, the latter nested inside the former, to avoid the undefined demand issue specific to bads

On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake

HIM-10 is required for kinetochore structure and function on Caenorhabditis elegans holocentric chromosomes.

Macromolecular structures called kinetochores attach and move chromosomes within the spindle during chromosome segregation. Using electron microscopy, we identified a structure on the holocentric mitotic and meiotic chromosomes of Caenorhabditis elegans that resembles the mammalian kinetochore. This structure faces the poles on mitotic chromosomes but encircles meiotic chromosomes. Worm kinetochores require the evolutionarily conserved HIM-10 protein for their structure and function. HIM-10 localizes to the kinetochores and mediates attachment of chromosomes to the spindle. Depletion of HIM-10 disrupts kinetochore structure, causes a failure of bipolar spindle attachment, and results in chromosome nondisjunction. HIM-10 is related to the Nuf2 kinetochore proteins conserved from yeast to humans. Thus, the extended kinetochores characteristic of C. elegans holocentric chromosomes provide a guide to the structure, molecular architecture, and function of conventional kinetochores

Finding fair and efficient allocations

We study the problem of fair division, where the goal is to allocate a set of items among a set of agents in a ``fair" manner. In particular, we focus on settings in which the items to be divided are either indivisible goods or divisible bads. Despite their practical significance, both these settings have been much less investigated than the divisible goods setting. In the first part of the dissertation, we focus on the fair division of indivisible goods. Our fairness criterion is envy-freeness up to any good (EFX). An allocation is EFX if no agent envies another agent following the removal of a single good from the other agent's bundle. Despite significant investment by the research community, the existence of EFX allocations remains open and is considered one of the most important open problems in fair division. In this thesis, we make significant progress on this question. First, we show that when agents have general valuations, we can determine an EFX allocation with a small number of unallocated goods (almost EFX allocation). Second, we demonstrate that when agents have structured valuations, we can determine an almost EFX allocation that is also efficient in terms of Nash welfare. Third, we prove that EFX allocations exist when there are three agents with additive valuations. Finally, we reduce the problem of finding improved guarantees on EFX allocations to a novel problem in extremal graph theory. In the second part of this dissertation, we turn to the fair division of divisible bads. Like in the setting of divisible goods, competitive equilibrium with equal incomes (CEEI) has emerged as the best mechanism for allocating divisible bads. However, neither a polynomial time algorithm nor any hardness result is known for the computation of CEEI with bads. We study the problem of dividing bads in the classic Arrow-Debreu setting (a setting that generalizes CEEI). We show that in sharp contrast to the Arrow-Debreu setting with goods, determining whether a competitive equilibrium exists, is NP-hard in the case of divisible bads. Furthermore, we prove the existence of equilibrium under a simple and natural sufficiency condition. Finally, we show that even on instances that satisfy this sufficiency condition, determining a competitive equilibrium is PPAD-hard. Thus, we settle the complexity of finding a competitive equilibrium in the Arrow-Debreu setting with divisible bads.Die Arbeit untersucht das Problem der gerechten Verteilung (fair division), welches zum Ziel hat, eine Menge von Gegenständen (items) einer Menge von Akteuren (agents) \zuzuordnen". Dabei liegt der Schwerpunkt der Arbeit auf Szenarien, in denen die zu verteilenden Gegenstände entweder unteilbare Güter (indivisible goods) oder teilbare Pflichten (divisible bads) sind. Trotz ihrer praktischen Relevanz haben diese Szenarien in der Forschung bislang bedeutend weniger Aufmerksamkeit erfahren als das Szenario mit teilbaren Gütern (divisible goods). Der erste Teil der Arbeit konzentriert sich auf die gerechte Verteilung unteilbarer Güter. Unser Gerechtigkeitskriterium ist Neid-Freiheit bis auf irgendein Gut (envy- freeness up to any good, EFX). Eine Zuordnung ist EFX, wenn kein Akteur einen anderen Akteur beneidet, nachdem ein einzelnes Gut aus dem Bündel des anderen Akteurs entfernt wurde. Die Existenz von EFX-Zuordnungen ist trotz ausgeprägter Bemühungen der Forschungsgemeinschaft ungeklärt und wird gemeinhin als eine der wichtigsten offenen Fragen des Feldes angesehen. Wir unternehmen wesentliche Schritte hin zu einer Klärung dieser Frage. Erstens zeigen wir, dass wir für Akteure mit allgemeinen Bewertungsfunktionen stets eine EFX-Zuordnung finden können, bei der nur eine kleine Anzahl von Gütern unallokiert bleibt (partielle EFX-Zuordnung, almost EFX allocation). Zweitens demonstrieren wir, dass wir für Akteure mit strukturierten Bewertungsfunktionen eine partielle EFX-Zuordnung bestimmen können, die zusätzlich effizient im Sinne der Nash-Wohlfahrtsfunktion ist. Drittens beweisen wir, dass EFX-Zuordnungen für drei Akteure mit additiven Bewertungsfunktionen immer existieren. Schließlich reduzieren wir das Problem, verbesserte Garantien für EFX-Zuordnungen zu finden, auf ein neuartiges Problem in der extremalen Graphentheorie. Der zweite Teil der Arbeit widmet sich der gerechten Verteilung teilbarer Pflichten. Wie im Szenario mit teilbaren Gütern hat sich auch hier das Wettbewerbsgleichgewicht bei gleichem Einkommen (competitive equilibrium with equal incomes, CEEI) als der beste Allokationsmechanismus zur Verteilung teilbarer Pflichten erwiesen. Gleichzeitig sind weder polynomielle Algorithmen noch Schwere-Resultate für die Berechnung von CEEI mit Pflichten bekannt. Die Arbeit untersucht das Problem der Verteilung von Pflichten im klassischen Arrow-Debreu-Modell (einer Generalisierung von CEEI). Wir zeigen, dass es NP-hart ist, zu entscheiden, ob es im Arrow-Debreu-Modell mit Pflichten ein Wettbewerbsgleichgewicht gibt { im scharfen Gegensatz zum Arrow-Debreu-Modell mit Gütern. Ferner beweisen wir die Existenz eines Gleichgewichts unter der Annahme einer einfachen und natürlichen hinreichenden Bedingung. Schließlich zeigen wir, dass die Bestimmung eines Wettbewerbsgleichgewichts sogar für Eingaben, die unsere hinreichende Bedingung erfüllen, PPAD-hart ist. Damit klären wir die Komplexität des Auffindens eines Wettbewerbsgleichgewichts im Arrow-Debreu-Modell mit teilbaren Pflichten