arXiv subjects

Mohammad Azharuddin Sanpui

Publications and source records attributed to Mohammad Azharuddin Sanpui.

7 recordsLinked to original sources

arXiv2026

Simultaneously Efficient Allocation of Indivisible Items Across Multiple Dimensions

Many allocation problems are intrinsically multidimensional, since an item may contribute differently to several criteria, and optimizing a single aggregate objective can hide severe losses in other dimensions. We study how much efficiency can be guaranteed simultaneously when indivisible items have multiple attributes. To this end, we introduce the \emph{multidimensional efficient allocation} (MDEA) model, where each agent has an additive valuation in each dimension, and investigate simultaneous efficiency under utilitarian social welfare (USW) and egalitarian social welfare (ESW). Our results reveal a sharp worst-case frontier. For exact efficiency, maximizing the number of dimensions attaining the USW optimum admits a $c/\ell$-approximation for every fixed constant $c$, and this dependence on the number $\ell$ of dimensions is essentially unavoidable; for ESW, even deciding whether two dimensions can be optimized simultaneously is NP-hard with binary valuations. For approximate simultaneous efficiency in every dimension, we identify a tight threshold of order $1/\ell$, showing that such guarantees always exist for both USW and ESW, while any asymptotically better dependence on $\ell$ is impossible, even for binary valuations. Finally, we introduce three natural multidimensional Pareto notions and characterize both their relationships and their computational complexity.

cs.GT↗

arXiv2025

Simultaneously Fair Allocation of Indivisible Items Across Multiple Dimensions

This paper explores the fair allocation of indivisible items in a multidimensional setting, motivated by the need to address fairness in complex environments where agents assess bundles according to multiple criteria. Such multidimensional settings are not merely of theoretical interest but are central to many real-world applications. For example, cloud computing resources are evaluated based on multiple criteria such as CPU cores, memory, and network bandwidth. In such cases, traditional one dimensional fairness notions fail to capture fairness across multiple attributes. To address these challenges, we study two relaxed variants of envy-freeness: weak simultaneously envy-free up to c goods (weak sEFc) and strong simultaneously envy-free up to c goods (strong sEFc), which accommodate the multidimensionality of agents' preferences. Under the weak notion, for every pair of agents and for each dimension, any perceived envy can be eliminated by removing, if necessary, a different set of goods from the envied agent's allocation. In contrast, the strong version requires selecting a single set of goods whose removal from the envied bundle simultaneously eliminates envy in every dimension. We provide upper and lower bounds on the relaxation parameter c that guarantee the existence of weak or strong sEFc allocations, where these bounds are independent of the total number of items. In addition, we present algorithms for checking whether a weak or strong sEFc allocation exists. Moreover, we establish NP-hardness results for checking the existence of weak sEF1 and strong sEF1 allocations.

cs.GT↗

arXiv2025

Contiguous Allocation of Indivisible Items on a Path

We study the problem of allocating indivisible items on a path among agents. The objective is to find a fair and efficient allocation in which each agent's bundle forms a contiguous block on the line. We say that an instance is \emph{$(a, b)$-sparse} if each agent values at most $a$ items positively and each item is valued positively by at most $b$ agents. We demonstrate that, even when the valuations are binary additive, deciding whether every item can be allocated to an agent who wants it is NP-complete for the $(4,3)$-sparse instances. Consequently, we provide two fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, with respect to the number of agents and the number of items. Additionally, we present a $2$-approximation algorithm for the special case when the valuations are binary additive, and the maximum utility is equal to the number of items. Also, we provide a $1/a$-approximation algorithm for the $(a,b)$-sparse instances. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least $2$ or at most $1$ is NP-complete for the $(6,3)$-sparse instances, even when the valuations are binary additive. We present a $1/a$-approximation algorithm for maximizing egalitarian social welfare for the $(a,b)$-sparse instances. Besides, we give two FPT algorithms for maximizing egalitarian social welfare in terms of the number of agents and the number of items. We also explore the case where the order of the blocks of items allocated to the agents is predetermined. In this case, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. However, we determine that checking the existence of an EF1 allocation is NP-complete, even when the valuations are binary additive.

cs.GT↗

arXiv2025

Resource Allocation under the Latin Square Constraint

A Latin square is an $n \times n$ matrix filled with $n$ distinct symbols, each of which appears exactly once in each row and exactly once in each column. We introduce a problem of allocating $n$ indivisible items among $n$ agents over $n$ rounds while satisfying the Latin square constraint. This constraint ensures that each agent receives no more than one item per round and receives each item at most once. Each agent has an additive valuation on the item--round pairs. Real-world applications like scheduling, resource management, and experimental design require the Latin square constraint to satisfy fairness or balancedness in allocation. Our goal is to find a partial or complete allocation that maximizes the sum of the agents' valuations (utilitarian social welfare) or the minimum of the agents' valuations (egalitarian social welfare). For the problem of maximizing utilitarian social welfare, we prove NP-hardness even when the valuations are binary additive. We then provide $(1-1/e)$ and $(1-1/e)/4$-approximation algorithms for partial and complete settings, respectively. Additionally, we present fixed-parameter tractable (FPT) algorithms with respect to the order of Latin square and the optimum value for both partial and complete settings. For the problem of maximizing egalitarian social welfare, we establish that deciding whether the optimum value is at most $1$ or at least $2$ is NP-hard for both the partial and complete settings, even when the valuations are binary. Furthermore, we demonstrate that checking the existence of a complete allocation that satisfies each of envy-free, proportional, equitable, envy-free up to any good, proportional up to any good, or equitable up to any good is NP-hard, even when the valuations are identical.

cs.GT↗

arXiv2024

Fair Division of Multi-layered Cakes

We consider multi-layered cake cutting in order to fairly allocate numerous divisible resources (layers of cake) among a group of agents under two constraints: contiguity and feasibility. We first introduce a new computational model in a multi-layered cake named ``a pair of knives''. Then, we show the existence of an exact multi-allocation for two agents and two layers using the new computational model. We demonstrate the computation procedure of a feasible and contiguous proportional multi-allocation over a three-layered cake for more than three agents. Finally, we develop a technique for computing proportional allocations for any number $n\geq 2^a3$ of agents and $2^a3$ layers, where $a$ is any positive integer.

cs.AI↗

arXiv2024

Externalities in Chore Division

The chore division problem simulates the fair division of a heterogeneous, undesirable resource among several agents. In the fair division of chores, each agent only gets the disutility from its own piece. Agents may, however, also be concerned with the pieces given to other agents; these externalities naturally appear in fair division situations. We first demonstrate the generalization of the classical concepts of proportionality and envy-freeness while extending the classical model by taking externalities into account.

cs.GT↗

arXiv2023

Chore Cutting: Envy and Truth

We study the fair division of divisible bad resources with strategic agents who can manipulate their private information to get a better allocation. Within certain constraints, we are particularly interested in whether truthful envy-free mechanisms exist over piecewise-constant valuations. We demonstrate that no deterministic truthful envy-free mechanism can exist in the connected-piece scenario, and the same impossibility result occurs for hungry agents. We also show that no deterministic, truthful dictatorship mechanism can satisfy the envy-free criterion, and the same result remains true for non-wasteful constraints rather than dictatorship. We further address several related problems and directions.

cs.GT↗