scholarly journals Guaranteeing Maximin Shares: Some Agents Left Behind

2021 ◽  
Author(s):  
Hadi Hosseini ◽  
Andrew Searns
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Synthetic Data ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Optimal Approximation ◽  
Constant Number ◽  
Indivisible Goods ◽  
Approximation Techniques ◽  
Left Behind

The maximin share (MMS) guarantee is a desirable fairness notion for allocating indivisible goods. While MMS allocations do not always exist, several approximation techniques have been developed to ensure that all agents receive a fraction of their maximin share. We focus on an alternative approximation notion, based on the population of agents, that seeks to guarantee MMS for a fraction of agents. We show that no optimal approximation algorithm can satisfy more than a constant number of agents, and discuss the existence and computation of MMS for all but one agent and its relation to approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for 2/3 of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee the value that an agent receives by partitioning the goods into 3n/2 bundles, improving the best known guarantee when goods are partitioned into 2n-2 bundles. Finally, we provide empirical experiments using synthetic data.

scholarly journals On the Max-Min Fair Stochastic Allocation of Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 2070-2078
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita
Keyword(s):  
Approximate Solution ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Indivisible Goods ◽  
Additive Utility ◽  
Ex Ante ◽  
Stochastic Setting

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.

scholarly journals Maximin-Aware Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Hau Chan ◽  
Jing Chen ◽  
Bo Li ◽  
Xiaowei Wu
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Indivisible Goods ◽  
Know How

We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].

OPTIMAL TRIANGULATIONS OF POINTS AND SEGMENTS WITH STEINER POINTS

2010 ◽  
Vol 20 (01) ◽  
pp. 89-104 ◽  
Author(s):  
BORIS ARONOV ◽  
TETSUO ASANO ◽  
STEFAN FUNKE
Keyword(s):  
Convex Hull ◽  
Polynomial Time ◽  
Natural Extension ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Optimal Location ◽  
Constant Number ◽  
Steiner Points ◽  
Vertex Set ◽  
Internal Angle

Consider a set X of points in the plane and a set E of non-crossing segments with endpoints in X. One can efficiently compute the triangulation of the convex hull of the points, which uses X as the vertex set, respects E, and maximizes the minimum internal angle of a triangle. In this paper we consider a natural extension of this problem: Given in addition a Steiner pointp, determine the optimal location of p and a triangulation of X ∪ {p} respecting E, which is best among all triangulations and placements of p in terms of maximizing the minimum internal angle of a triangle. We present a polynomial-time algorithm for this problem and then extend our solution to handle any constant number of Steiner points.

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

An improved primal-dual approximation algorithm for the k-means problem with penalties

2021 ◽  
pp. 1-13
Author(s):  
Chunying Ren ◽  
Dachuan Xu ◽  
Donglei Du ◽  
Min Li
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Best Approximation ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Ratio ◽  
Data Set ◽  
Penalty Cost ◽  
Primal Dual ◽  
Total Penalty

Abstract In the k-means problem with penalties, we are given a data set ${\cal D} \subseteq \mathbb{R}^\ell $ of n points where each point $j \in {\cal D}$ is associated with a penalty cost p j and an integer k. The goal is to choose a set ${\rm{C}}S \subseteq {{\cal R}^\ell }$ with |CS| ≤ k and a penalized subset ${{\cal D}_p} \subseteq {\cal D}$ to minimize the sum of the total squared distance from the points in D / D p to CS and the total penalty cost of points in D p , namely $\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$ . We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

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 A POLYNOMIAL-TIME APPROXIMATION ALGORITHM FOR A GEOMETRIC DISPERSION PROBLEM

2009 ◽  
Vol 19 (03) ◽  
pp. 267-288 ◽  
Author(s):  
MARC BENKERT ◽  
JOACHIM GUDMUNDSSON ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
ALEXANDER WOLFF
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Packing Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Dispersion Problem ◽  
Unit Disks ◽  
Polynomial Time Approximation

We consider the following packing problem. Let α be a fixed real in (0, 1]. We are given a bounding rectangle ρ and a set [Formula: see text] of n possibly intersecting unit disks whose centers lie in ρ. The task is to pack a set [Formula: see text] of m disjoint disks of radius α into ρ such that no disk in B intersects a disk in [Formula: see text], where m is the maximum number of unit disks that can be packed. In this paper we present a polynomial-time algorithm for α = 2/3. So far only the case of packing squares has been considered. For that case, Baur and Fekete have given a polynomial-time algorithm for α = 2/3 and have shown that the problem cannot be solved in polynomial time for any α > 13/14 unless [Formula: see text].

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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