scholarly journals Fair Pairwise Exchange among Groups

2021 ◽  
Author(s):  
Zhaohong Sun ◽  
Taiki Todo ◽  
Toby Walsh
Keyword(s):  
Polynomial Time ◽  
Real World ◽  
Research Question ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Individual Rationality ◽  
Running Time ◽  
Graph Structures ◽  
Real World Applications

We study the pairwise organ exchange problem among groups motivated by real-world applications and consider two types of group formulations. Each group represents either a certain type of patient-donor pairs who are compatible with the same set of organs, or a set of patient-donor pairs who reside in the same region. We address a natural research question, which asks how to match a maximum number of pairwise compatible patient-donor pairs in a fair and individually rational way. We first propose a natural fairness concept that is applicable to both types of group formulations and design a polynomial-time algorithm that checks whether a matching exists that satisfies optimality, individual rationality, and fairness. We also present several running time upper bounds for computing such matchings for different graph structures.

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 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.

scholarly journals An expected polynomial time algorithm for coloring 2-colorable 3-graphs

2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Yury Person ◽  
Mathias Schacht
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Running Time ◽  
Uniform Hypergraphs ◽  
Average Running Time ◽  
International Audience

Graphs and Algorithms International audience We present an algorithm that for 2-colorable 3-uniform hypergraphs, finds a 2-coloring in average running time O (n(5) log(2) n).

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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