arXiv subjects

Ryoga Mahara

Publications and source records attributed to Ryoga Mahara.

10 recordsLinked to original sources

arXiv2026

Fair and Efficient Balanced Allocation for Indivisible Goods

We study the problem of allocating indivisible goods among agents with additive valuation functions to achieve both fairness and efficiency under the constraint that each agent receives exactly the same number of goods (the \emph{balanced constraint}). While this constraint is common in real-world scenarios such as team drafts or asset division, it significantly complicates the search for allocations that are both fair and efficient. Envy-freeness up to one good (EF1) is a well-established fairness notion for indivisible goods. Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely accepted efficiency criteria. Our main contribution establishes both the existence and polynomial-time computability of allocations that are simultaneously EF1 and fPO under balanced constraints in two fundamental cases: (1) when each agent has a personalized bivalued valuation, and (2) when agents have at most two distinct valuation types,. Our algorithms leverage novel applications of maximum-weight matching in bipartite graphs and duality theory, providing the first polynomial-time solutions for these cases and offering new insights for constrained fair division problems.

cs.GT↗

arXiv2026

The Power of Share-Based Notions in Proving Envy-Based Fairness Guarantees

We study the problem of fairly allocating indivisible goods among agents with monotone valuations. We introduce a new share-based fairness notion, the residual maximin share (RMMS), and show that it provides a unified framework for several existing lone-divider style techniques in fair division. RMMS satisfies two key properties: feasibility and self-maximization. Using RMMS, we give simple proofs of the existence of partial allocations that are both RMMS and envy-free up to any good (EFX), and complete allocations that are both RMMS and envy-free up to one good (EF1), in fact satisfying the stronger notion of EFL. This unifies and strengthens several previously known results. We further demonstrate the power of the share-based approach by studying the compatibility of fairness notions related to the long-standing EFX problem. While allocations satisfying either epistemic EFX (EEFX) or EF1 are known to exist for general monotone valuations, whether they can always be achieved simultaneously has remained open in every setting where EFX existence itself is unresolved. For additive valuations, we resolve this question affirmatively by proving the existence of allocations that satisfy both EEFX and EFL. Our proof introduces the strong EEFX share, a new share notion implying EEFX feasibility of bundles. We show that the strong EEFX share is upper bounded by RMMS, enabling us to derive EEFX+EFL allocations via the RMMS framework. This answers the main open question of Akrami and Rathi (2025). Finally, although our algorithm for computing EEFX and EF1 allocations may take exponential time in general, we develop a polynomial-time algorithm for restricted additive valuations. Unlike the lone-divider approach, our algorithm exploits the structural properties of restricted additive valuations to compute allocations satisfying both EEFX and EF1.

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arXiv2025

Existence of Fair and Efficient Allocation of Indivisible Chores

We study the problem of allocating indivisible chores among agents with additive cost functions in a fair and efficient manner. A major open question in this area is whether there always exists an allocation that is envy-free up to one chore (EF1) and Pareto optimal (PO). Our main contribution is to provide a positive answer to this question by proving the existence of such an allocation for indivisible chores under additive cost functions. This is achieved by a novel combination of a fixed point argument and a discrete algorithm, providing a significant methodological advance in this area. Our additional key contributions are as follows. We show that there always exists an allocation that is EF1 and fractional Pareto optimal (fPO), where fPO is a stronger efficiency concept than PO. We also show that an EF1 and PO allocation can be computed in polynomial time when the number of agents is constant. Finally, we extend all of these results to the more general setting of weighted EF1 (wEF1), which accounts for the entitlements of agents.

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arXiv2024

A Polynomial-Time Algorithm for Fair and Efficient Allocation with a Fixed Number of Agents

We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuation functions. Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods, while Pareto optimality (PO) and its stronger variant, fractional Pareto optimality (fPO), are widely recognized efficiency criteria. Although each property is straightforward to achieve individually, simultaneously ensuring both fairness and efficiency is challenging. Caragiannis et al.~\cite{caragiannis2019unreasonable} established the surprising result that maximizing Nash social welfare yields an allocation that is both EF1 and PO; however, since maximizing Nash social welfare is NP-hard, this approach does not provide an efficient algorithm. To overcome this barrier, Barman, Krishnamurthy, and Vaish~\cite{barman2018finding} designed a pseudo-polynomial time algorithm to compute an EF1 and PO allocation, and showed the existence of EF1 and fPO allocations. Nevertheless, the latter existence proof relies on a non-constructive convergence argument and does not directly yield an efficient algorithm for finding EF1 and fPO allocations. Whether a polynomial-time algorithm exists for finding an EF1 and PO (or fPO) allocation remains an important open problem. In this paper, we propose a polynomial-time algorithm to compute an allocation that achieves both EF1 and fPO under additive valuation functions when the number of agents is fixed. Our primary idea is to avoid processing the entire instance at once; instead, we sequentially add agents to the instance and construct an allocation that satisfies EF1 and fPO at each step.

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arXiv2024

Position Fair Mechanisms Allocating Indivisible Goods

Fair division mechanisms for indivisible goods require agent orderings to deterministically select one allocation when running the algorithm in practice. We introduce position envy-freeness up to one good (PEF1) as a fairness criterion for mechanisms: a mechanism is said to satisfy PEF1 if for any pair of agent orderings, no agent prefers their bundle determined under one ordering to that under another ordering by more than the utility of a single good. First, we propose a scale-invariant, polynomial-time mechanism that satisfies PEF1 and yields an envy-freeness up to one good (EF1) allocation. For the case of two agents, we establish that any mechanism producing a maximum Nash welfare allocation eliminates envy based on positions by removing one good, provided that utilities are positive. Additionally, we present a polynomial-time mechanism based on the adjusted winner procedure, which satisfies PEF1 and produces an EF1 and Pareto optimal allocation for two agents. In contrast, we demonstrate that well-known mechanisms such as round-robin and envy-cycle elimination do not generally satisfy PEF1.

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arXiv2023

Reconfiguration of the Union of Arborescences

An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal the reconfigurability of the union of $k$ arborescences for fixed $k$ in the following sense: for any pair of arc subsets that can be partitioned into $k$ arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into $k$ arborescences. This generalizes the result by Ito et al. (2023), who showed the case with $k=1$. Since the union of $k$ arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

cs.DM↗

arXiv2023

EFX Allocations for Indivisible Chores: Matching-Based Approach

One of the most important topics in discrete fair division is whether an EFX allocation exists for any instance. Although the existence of EFX allocations is a standing open problem for both goods and chores, the understanding of the existence of EFX allocations for chores is less established compared to goods. We study the existence of EFX allocation for chores under the assumption that all agent's cost functions are additive. Specifically, we show the existence of EFX allocations for the following three cases: (i) the number of chores is at most twice the number of agents, (ii) the cost functions of all agents except for one are identical ordering, and (iii) the number of agents is three and each agent has a personalized bi-valued cost function. Furthermore, we provide a polynomial time algorithm to find an EFX allocation for each case.

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arXiv2023

Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average

We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate concepts of proportionality have been considered in the previous work. Among them, proportionality up to the maximin good (PROPm) has been the best approximate notion of proportionality that can be achieved for all instances. In this paper, we introduce the notion of proportionality up to the least valued good on average (PROPavg), which is a stronger notion than PROPm, and show that a PROPavg allocation always exists for all instances and can be computed in polynomial time. %% for all instances. Our results establish PROPavg as a notable non-trivial fairness notion that can be achieved for all instances. Our proof is constructive, and based on a new technique that generalizes the cut-and-choose protocol and uses a recursive technique.

cs.GT↗

arXiv2021

Extension of Additive Valuations to General Valuations on the Existence of EFX

Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when $n=2$ or (ii) when all agents have identical valuations, where $n$ is the number of agents. it is also known that an EFX allocation always exists when one can leave at most $n-1$ items unallocated. We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most $n+3$. We also show that an EFX allocation always exists when one can leave at most $n-2$ items unallocated. In addition to the positive results, we construct an instance with $n=3$ in which an existing approach does not work as it is.

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arXiv2020

Existence of EFX for Two Additive Valuations

Fair division of indivisible items is a well-studied topic in Economics and Computer Science. The objective is to allocate items to agents in a fair manner, where each agent has a valuation for each subset of items. Envy-freeness is one of the most widely studied notions of fairness. Since complete envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling one is envy-freeness up to any item (EFX), where no agent envies another agent after the removal of any single item from the other agent's bundle. However, despite significant efforts by many researchers for several years, it is known that a complete EFX allocation always exists only in limited cases. In this paper, we show that a complete EFX allocation always exists when each agent is of one of two given types, where agents of the same type have identical additive valuations. This is the first such existence result for non-identical valuations when there are any number of agents and items and no limit on the number of distinct values an agent can have for individual items. We give a constructive proof, in which we iteratively obtain a Pareto dominating (partial) EFX allocation from an existing partial EFX allocation.

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