scholarly journals Almost Group Envy-free Allocation of Indivisible Goods and Chores

2020 ◽  
Author(s):  
Haris Aziz ◽  
Simon Rey
Keyword(s):  
Resource Allocation ◽  
Polynomial Time ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Preference Domain ◽  
Polynomial Time Algorithms ◽  
Multi Agent ◽  
Additive Utilities ◽  
Natural Classes ◽  
Allocation Of Indivisible Items

We consider a multi-agent resource allocation setting in which an agent's utility may decrease or increase when an item is allocated. We take the group envy-freeness concept that is well-established in the literature and present stronger and relaxed versions that are especially suitable for the allocation of indivisible items. Of particular interest is a concept called group envy-freeness up to one item (GEF1). We then present a clear taxonomy of the fairness concepts. We study which fairness concepts guarantee the existence of a fair allocation under which preference domain. For two natural classes of additive utilities, we design polynomial-time algorithms to compute a GEF1 allocation. We also prove that checking whether a given allocation satisfies GEF1 is coNP-complete when there are either only goods, only chores or both.

scholarly journals Solving Multi-agent Path Finding on Strongly Biconnected Digraphs

2018 ◽  
Vol 62 ◽  
pp. 273-314
Author(s):  
Adi Botea ◽  
Davide Bonusi ◽  
Pavel Surynek
Keyword(s):  
Empirical Analysis ◽  
Polynomial Time ◽  
Directed Graphs ◽  
Path Finding ◽  
Undirected Graphs ◽  
Suboptimal Solutions ◽  
Polynomial Time Algorithms ◽  
Asymmetric Communication ◽  
Multi Agent

Much of the literature on suboptimal, polynomial-time algorithms for multi-agent path finding focuses on undirected graphs, where motion is permitted in both directions along a graph edge. Despite this, traveling on directed graphs is relevant in navigation domains, such as path finding in games, and asymmetric communication networks.We consider multi-agent path finding on strongly biconnected directed graphs. We show that all instances with at least two unoccupied positions have a solution, except for a particular, degenerate subclass where the graph has a cyclic shape. We present diBOX, an algorithm for multi-agent path finding on strongly biconnected directed graphs. diBOX runs in polynomial time, computes suboptimal solutions and is complete for instances on strongly biconnected digraphs with at least two unoccupied positions. We theoretically analyze properties of the algorithm and properties of strongly biconnected directed graphs that are relevant to our approach. We perform a detailed empirical analysis of diBOX, showing a good scalability. To our knowledge, our work is the first study of multi-agent path finding focused on directed graphs.

scholarly journals On the Proximity of Markets with Integral Equilibria

2019 ◽  
Vol 33 ◽  
pp. 1748-1755 ◽  
Author(s):  
Siddharth Barman ◽  
Sanath Kumar Krishnamurthy
Keyword(s):  
Polynomial Time ◽  
Fair Division ◽  
Prior Work ◽  
Indivisible Goods ◽  
Polynomial Time Algorithms ◽  
Fairness And Efficiency ◽  
Pareto Efficient ◽  
Efficient Allocations ◽  
Strong Existence ◽  
Strongly Polynomial

We study Fisher markets that admit equilibria wherein each good is integrally assigned to some agent. While strong existence and computational guarantees are known for equilibria of Fisher markets with additive valuations (Eisenberg and Gale 1959; Orlin 2010), such equilibria, in general, assign goods fractionally to agents. Hence, Fisher markets are not directly applicable in the context of indivisible goods. In this work we show that one can always bypass this hurdle and, up to a bounded change in agents’ budgets, obtain markets that admit an integral equilibrium. We refer to such markets as pure markets and show that, for any given Fisher market (with additive valuations), one can efficiently compute a “near-by,” pure market with an accompanying integral equilibrium.Our work on pure markets leads to novel algorithmic results for fair division of indivisible goods. Prior work in discrete fair division has shown that, under additive valuations, there always exist allocations that simultaneously achieve the seemingly incompatible properties of fairness and efficiency (Caragiannis et al. 2016); here fairness refers to envyfreeness up to one good (EF1) and efficiency corresponds to Pareto efficiency. However, polynomial-time algorithms are not known for finding such allocations. Considering relaxations of proportionality and EF1, respectively, as our notions of fairness, we show that fair and Pareto efficient allocations can be computed in strongly polynomial time.

scholarly journals Fair Division Under Cardinality Constraints

2018 ◽  
Author(s):  
Arpita Biswas ◽  
Siddharth Barman
Keyword(s):  
Resource Allocation ◽  
Fair Division ◽  
Division Problem ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Solution Concepts ◽  
Cardinality Constraints ◽  
The Given

We consider the problem of fairly allocating indivisible goods, among agents, under cardinality constraints and additive valuations. In this setting, we are given a partition of the entire set of goods---i.e., the goods are categorized---and a limit is specified on the number of goods that can be allocated from each category to any agent. The objective here is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. This problem naturally captures a number of resource-allocation applications, and is a generalization of the well-studied unconstrained fair division problem.  The two central notions of fairness, in the context of fair division of indivisible goods, are envy freeness up to one good (EF1) and the (approximate) maximin share guarantee (MMS). We show that the existence and algorithmic guarantees established for these solution concepts in the unconstrained setting can essentially be achieved under cardinality constraints. Furthermore, focusing on the case wherein all the agents have the same additive valuation, we establish that EF1 allocations exist even under matroid constraints.

scholarly journals Maximin Share Allocations on Cycles

2020 ◽  
Vol 69 ◽  
pp. 613-655
Author(s):  
Miroslaw Truszczynski ◽  
Zbigniew Lonc
Keyword(s):  
Resource Allocation ◽  
Social Choice ◽  
Fundamental Problem ◽  
Fair Division ◽  
Connected Subgraph ◽  
Indivisible Goods ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Multi Agent ◽  
Share Value

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

scholarly journals Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

scholarly journals Almost Envy-Freeness in Group Resource Allocation

2019 ◽  
Author(s):  
Maria Kyropoulou ◽  
Warut Suksompong ◽  
Alexandros A. Voudouris
Keyword(s):  
Resource Allocation ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
New Model

We study the problem of fairly allocating indivisible goods between groups of agents using the recently introduced relaxations of envy-freeness. We consider the existence of fair allocations under different assumptions on the valuations of the agents. In particular, our results cover cases of arbitrary monotonic, responsive, and additive valuations, while for the case of binary valuations we fully characterize the cardinalities of two groups of agents for which a fair allocation can be guaranteed with respect to both envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). Moreover, we introduce a new model where the agents are not partitioned into groups in advance, but instead the partition can be chosen in conjunction with the allocation of the goods. In this model, we show that for agents with arbitrary monotonic valuations, there is always a partition of the agents into two groups of any given sizes along with an EF1 allocation of the goods. We also provide an extension of this result to any number of groups.

scholarly journals A Complexity Approach for Core-Selecting Exchange under Conditionally Lexicographic Preferences

2018 ◽  
Vol 63 ◽  
pp. 515-555 ◽  
Author(s):  
Etsushi Fujita ◽  
Julien Lesca ◽  
Akihisa Sonoda ◽  
Taiki Todo ◽  
Makoto Yokoo
Keyword(s):  
Resource Allocation ◽  
Mechanism Design ◽  
Polynomial Time ◽  
Indivisible Goods ◽  
Lexicographic Preferences ◽  
Side Payments ◽  
Different Types ◽  
Core Selection ◽  
Exchange Rule ◽  
Relationship Of

Core-selection is a crucial property of rules in the literature of resource allocation. It is also desirable, from the perspective of mechanism design, to address the incentive of agents to cheat by misreporting their preferences. This paper investigates the exchange problem where (i) each agent is initially endowed with (possibly multiple) indivisible goods, (ii) agents' preferences are assumed to be conditionally lexicographic, and (iii) side payments are prohibited. We propose an exchange rule called augmented top-trading-cycles (ATTC), based on the original TTC procedure. We first show that ATTC is core-selecting and runs in polynomial time with respect to the number of goods. We then show that finding a beneficial misreport under ATTC is NP-hard. We finally clarify relationship of misreporting with splitting and hiding, two different types of manipulations, under ATTC.

scholarly journals Fairness Towards Groups of Agents in the Allocation of Indivisible Items

2019 ◽  
Author(s):  
Nawal Benabbou ◽  
Mithun Chakraborty ◽  
Edith Elkind ◽  
Yair Zick
Keyword(s):  
Social Welfare ◽  
Polynomial Time ◽  
Utility Maximization ◽  
Marginal Utility ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Indivisible Goods ◽  
Allocation Of Indivisible Items

In this paper, we study the problem of matching a set of items to a set of agents partitioned into types so as to balance fairness towards the types against overall utility/efficiency. We extend multiple desirable properties of indivisible goods allocation to our model and investigate the possibility and hardness of achieving combinations of these properties, e.g. we prove that maximizing utilitarian social welfare under constraints of typewise envy-freeness up to one item (TEF1) is computationally intractable. We also define a new concept of waste for this setting, show experimentally that augmenting an existing algorithm with a marginal utility maximization heuristic can produce a TEF1 solution with reduced waste, and also provide a polynomial-time algorithm for computing a non-wasteful TEF1 allocation for binary agent-item utilities.

scholarly journals Fair allocation of conflicting items

2021 ◽  
Vol 36 (1) ◽  
Author(s):  
Halvard Hummel ◽  
Magnus Lie Hetland
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Conflict Graph ◽  
Fair Allocation ◽  
Conflicting Interests ◽  
To Receive ◽  
Allocation Of Indivisible Items

AbstractWe study fair allocation of indivisible items, where the items are furnished with a set of conflicts, and agents are not permitted to receive conflicting items. This kind of constraint captures, for example, participating in events that overlap in time, or taking on roles in the presence of conflicting interests. We demonstrate, both theoretically and experimentally, that fairness characterizations such as EF1, MMS and MNW still are applicable and useful under item conflicts. Among other existence, non-existence and computability results, we show that a $$1/\Delta $$ 1 / Δ -approximate MMS allocation for n agents may be found in polynomial time when $$n>\Delta >2$$ n > Δ > 2 , for any conflict graph with maximum degree $$\Delta$$ Δ , and that, if $$n > \Delta $$ n > Δ , a 1/3-approximate MMS allocation always exists.

A Game-Theoretic Formulation of Multi-Agent Resource Allocation

2005 ◽  
Author(s):  
Jonathan Bredin ◽  
Rajiv T. Maheswaran ◽  
Cagri Imer ◽  
Tamer Basar ◽  
David Kotz ◽  
...  
Keyword(s):  
Resource Allocation ◽  
Multi Agent ◽  
Game Theoretic
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