arXiv subjects

Hirota Kinoshita

Publications and source records attributed to Hirota Kinoshita.

5 recordsLinked to original sources

arXiv2026

Maximin Shares with Lower Quotas

We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to lower and upper quotas on the number of items allocated to each agent. Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution. This paper focuses on the fairness criterion known as maximin shares (MMS) and its approximations. Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores. Furthermore, we consider the generalized scenario where items are partitioned into multiple categories, each with its own lower and upper quotas. In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time. These results extend previous work on the cardinality constraints, i.e., the special case where only upper quotas are imposed.

cs.GT↗

arXiv2026

A Relative-Budget Theory for Reinforcement Learning with Verifiable Rewards in Large Language Model Reasoning

Reinforcement learning (RL) is a dominant paradigm for improving the reasoning abilities of large language models, yet its effectiveness varies across tasks and compute budgets. We propose a \emph{relative-budget} theory explaining this variation through a single quantity called relative budget $ξ:= H/\mathbb{E}[T]$, where $H$ is the generation horizon (token budget) and $T$ denotes the number of tokens until the first correct solution under a base policy. We show that $ξ$ determines sample efficiency by controlling reward variance and the likelihood of informative trajectories. Our analysis reveals three regimes: in the \emph{deficient} regime ($ξ\to 0$), informative trajectories are rare and the sample complexity explodes; in the \emph{balanced} regime ($ξ=Θ(1)$), informative trajectories occur with non-negligible probability and RL is maximally sample-efficient; and in the \emph{ample} regime ($ξ\to \infty$), learning remains stable but marginal gains per iteration diminish. We further provide finite-sample guarantees for online RL that characterize learning progress across these regimes. Specifically, in a case study under idealized distributional assumptions, we show that the relative budget grows linearly over iterations. Our empirical results confirm these predictions in realistic settings, identifying a budget $ξ\in [1.5, 2.0]$ that maximizes learning efficiency and coincides with peak reasoning performance.

cs.LG↗

arXiv2025

Achieving PAC Guarantees in Mechanism Design through Multi-Armed Bandits

We analytically derive a class of optimal solutions to a linear program (LP) for automated mechanism design that satisfies efficiency, incentive compatibility, strong budget balance (SBB), and individual rationality (IR), where SBB and IR are enforced in expectation. These solutions can be expressed using a set of essential variables whose cardinality is exponentially smaller than the total number of variables in the original formulation. However, evaluating a key term in the solutions requires exponentially many optimization steps as the number of players $N$ increases. We address this by translating the evaluation of this term into a multi-armed bandit (MAB) problem and develop a probably approximately correct (PAC) estimator with asymptotically optimal sample complexity. This MAB-based approach reduces the optimization complexity from exponential to $O(N\log N)$. Numerical experiments confirm that our method efficiently computes mechanisms with the target properties, scaling to problems with up to $N=128$ players -- substantially improving over prior work.

cs.GT↗

arXiv2024

A Faster Deterministic Algorithm for Mader's $\mathcal{S}$-Path Packing

Given an undirected graph $G = (V,E)$ with a set of terminals $T\subseteq V$ partitioned into a family $\mathcal{S}$ of disjoint blocks, find the maximum number of vertex-disjoint paths whose endpoints belong to two distinct blocks while no other internal vertex is a terminal. This problem is called Mader's $\mathcal{S}$-path packing. It has been of remarkable interest as a common generalization of the non-bipartite matching and vertex-disjoint $s\text{-}t$ paths problem. This paper presents a new deterministic algorithm for this problem via known reduction to linear matroid parity. The algorithm utilizes the augmenting-path algorithm of Gabow and Stallmann (1986), while replacing costly matrix operations between augmentation steps with a faster algorithm that exploits the original $\mathcal{S}$-path packing instance. The proposed algorithm runs in $O(mnk)$ time, where $n = |V|$, $m = |E|$, and $k = |T|\le n$. This improves on the previous best bound $O(mn^ω)$ for deterministic algorithms, where $ω\ge2$ denotes the matrix multiplication exponent.

cs.DS↗

arXiv2024

Socially efficient mechanism on the minimum budget

In social decision-making among strategic agents, a universal focus lies on the balance between social and individual interests. Socially efficient mechanisms are thus desirably designed to not only maximize the social welfare but also incentivize the agents for their own profit. Under a generalized model that includes applications such as double auctions and trading networks, this study establishes a socially efficient (SE), dominant-strategy incentive compatible (DSIC), and individually rational (IR) mechanism with the minimum total budget expensed to the agents. The present method exploits discrete and known type domains to reduce a set of constraints into the shortest path problem in a weighted graph. In addition to theoretical derivation, we substantiate the optimality of the proposed mechanism through numerical experiments, where it certifies strictly lower budget than Vickery-Clarke-Groves (VCG) mechanisms for a wide class of instances.

cs.GT↗