scholarly journals The Fair Division of Hereditary Set Systems

2021 ◽  
Vol 9 (2) ◽  
pp. 1-19
Author(s):  
Z. Li ◽  
A. Vetta
Keyword(s):  
Recent Work ◽  
Polynomial Time ◽  
Simple Algorithm ◽  
Fair Division ◽  
Set Systems

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form a hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3666 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

scholarly journals Multiple Birds with One Stone: Beating 1/2 for EFX and GMMS via Envy Cycle Elimination

2020 ◽  
Vol 34 (02) ◽  
pp. 1790-1797 ◽  
Author(s):  
Georgios Amanatidis ◽  
Evangelos Markakis ◽  
Apostolos Ntokos
Keyword(s):  
Polynomial Time ◽  
Golden Ratio ◽  
Simple Algorithm ◽  
Fair Division ◽  
Existence Results ◽  
Indivisible Goods ◽  
Approximation Factor ◽  
General Existence ◽  
Primary Focus ◽  
Fine Tune

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ−1)-approximation of envy-freeness up to any good (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS.Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.

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 Weighted Envy-freeness in Indivisible Item Allocation

2021 ◽  
Vol 9 (3) ◽  
pp. 1-39
Author(s):  
Mithun Chakraborty ◽  
Ayumi Igarashi ◽  
Warut Suksompong ◽  
Yair Zick
Keyword(s):  
Social Welfare ◽  
Polynomial Time ◽  
Pareto Optimality ◽  
Arbitrary Number ◽  
Fair Division ◽  
Pareto Optimal ◽  
Weak Version ◽  
Strong Version ◽  
Two Agents ◽  
Allocation Of Indivisible Items

We introduce and analyze new envy-based fairness concepts for agents with weights that quantify their entitlements in the allocation of indivisible items. We propose two variants of weighted envy-freeness up to one item (WEF1): strong , where envy can be eliminated by removing an item from the envied agent’s bundle, and weak , where envy can be eliminated either by removing an item (as in the strong version) or by replicating an item from the envied agent’s bundle in the envying agent’s bundle. We show that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1, but it always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence algorithm produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can efficiently achieve both strong WEF1 and Pareto optimality by adapting the adjusted winner procedure. Our work highlights several aspects in which weighted fair division is richer and more challenging than its unweighted counterpart.

scholarly journals Combining the Delete Relaxation with Critical-Path Heuristics: A Direct Characterization

2016 ◽  
Vol 56 ◽  
pp. 269-327 ◽  
Author(s):  
Maximilian Fickert ◽  
Joerg Hoffmann ◽  
Marcel Steinmetz
Keyword(s):  
Recent Work ◽  
Polynomial Time ◽  
Critical Path ◽  
Complexity Reduction ◽  
Information Loss ◽  
The Other ◽  
Other Hand ◽  
Np Complete ◽  
Planning Task ◽  
The One

Recent work has shown how to improve delete relaxation heuristics by computing relaxed plans, i.e., the hFF heuristic, in a compiled planning task PiC which represents a given set C of fact conjunctions explicitly. While this compilation view of such partial delete relaxation is simple and elegant, its meaning with respect to the original planning task is opaque, and the size of PiC grows exponentially in |C|. We herein provide a direct characterization, without compilation, making explicit how the approach arises from a combination of the delete-relaxation with critical-path heuristics. Designing equations characterizing a novel view on h+ on the one hand, and a generalized version hC of hm on the other hand, we show that h+(PiC) can be characterized in terms of a combined hcplus equation. This naturally generalizes the standard delete-relaxation framework: understanding that framework as a relaxation over singleton facts as atomic subgoals, one can refine the relaxation by using the conjunctions C as atomic subgoals instead. Thanks to this explicit view, we identify the precise source of complexity in hFF(PiC), namely maximization of sets of supported atomic subgoals during relaxed plan extraction, which is easy for singleton-fact subgoals but is NP-complete in the general case. Approximating that problem greedily, we obtain a polynomial-time hCFF version of hFF(PiC), superseding the PiC compilation, and superseding the modified PiCce compilation which achieves the same complexity reduction but at an information loss. Experiments on IPC benchmarks show that these theoretical advantages can translate into empirical ones.

scholarly journals On the Performance of a Simple Approximation Algorithm for the Longest Path Problem

2019 ◽  
Vol 35 (1) ◽  
pp. 57-68
Author(s):  
Nguyen Thi Phuong ◽  
Tran Vinh Duc ◽  
Le Cong Thanh
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Undirected Graph ◽  
Simple Algorithm ◽  
Performance Ratio ◽  
Constant Ratio ◽  
Time Approximation ◽  
Longest Path ◽  
Longest Path Problem

The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\sqrt{|V| \log |V|}$ and so has performance ratio less than $1+4\sqrt{\log |V|/|V|}$.\

ON THE LIMIT CYCLE STRUCTURE OF THRESHOLD BOOLEAN NETWORKS OVER COMPLETE GRAPHS

2004 ◽  
Vol 14 (03) ◽  
pp. 209-215 ◽  
Author(s):  
GEORGE VOUTSADAKIS
Keyword(s):  
Fixed Points ◽  
Limit Cycle ◽  
Polynomial Time ◽  
Simple Algorithm ◽  
Boolean Networks ◽  
Complete Graphs ◽  
Cycle Structure ◽  
Arbitrary Length ◽  
Limit Structure

In previous work, the limit structure of positive and negative finite threshold boolean networks without inputs (TBNs) over the complete digraph Kn was analyzed and an algorithm was presented for computing this structure in polynomial time. Those results are generalized in this paper to cover the case of arbitrary TBNs over Kn. Although the limit structure is now more complicated, containing, not only fixed-points and cycles of length 2, but possibly also cycles of arbitrary length, a simple algorithm is still available for its determination in polynomial time. Finally, the algorithm is generalized to cover the case of symmetric finite boolean networks over Kn.

scholarly journals Group Fairness for the Allocation of Indivisible Goods

2019 ◽  
Vol 33 ◽  
pp. 1853-1860 ◽  
Author(s):  
Vincent Conitzer ◽  
Rupert Freeman ◽  
Nisarg Shah ◽  
Jennifer Wortman Vaughan
Keyword(s):  
Local Search ◽  
Polynomial Time ◽  
Fair Division ◽  
Indivisible Goods ◽  
Local Optima ◽  
Pseudo Polynomial Time

We consider the problem of fairly dividing a collection of indivisible goods among a set of players. Much of the existing literature on fair division focuses on notions of individual fairness. For instance, envy-freeness requires that no player prefer the set of goods allocated to another player to her own allocation. We observe that an algorithm satisfying such individual fairness notions can still treat groups of players unfairly, with one group desiring the goods allocated to another. Our main contribution is a notion of group fairness, which implies most existing notions of individual fairness. Group fairness (like individual fairness) cannot be satisfied exactly with indivisible goods. Thus, we introduce two “up to one good” style relaxations. We show that, somewhat surprisingly, certain local optima of the Nash welfare function satisfy both relaxations and can be computed in pseudo-polynomial time by local search. Our experiments reveal faster computation and stronger fairness guarantees in practice.

On a Problem of Danzer

2018 ◽  
Vol 28 (3) ◽  
pp. 473-482
Author(s):  
NABIL H. MUSTAFA ◽  
SAURABH RAY
Keyword(s):  
Convex Hull ◽  
Recent Work ◽  
Polynomial Time ◽  
Convex Sets ◽  
General Theorem ◽  
Convex Hulls ◽  
Exponential Bound ◽  
Convex Object ◽  
Polynomial Bounds ◽  
Bounded Convex

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

scholarly journals A Simple Algorithm for Optimal Search Trees with Two-way Comparisons

2022 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Marek Chrobak ◽  
Mordecai Golin ◽  
J. Ian Munro ◽  
Neal E. Young
Keyword(s):  
Polynomial Time ◽  
Simple Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Search Trees ◽  
Optimal Search ◽  
Correctness Proof ◽  
Structural Theorem ◽  
Running Time ◽  
Previous Solution

We present a simple O(n 4 ) -time algorithm for computing optimal search trees with two-way comparisons. The only previous solution to this problem, by Anderson et al., has the same running time but is significantly more complicated and is restricted to the variant where only successful queries are allowed. Our algorithm extends directly to solve the standard full variant of the problem, which also allows unsuccessful queries and for which no polynomial-time algorithm was previously known. The correctness proof of our algorithm relies on a new structural theorem for two-way-comparison search trees.

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