Dividing Splittable Goods Evenly and With Limited Fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases F = 1 and F = 2 . For F = 1 , the max–min and min–max problems are solvable in...

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Vydáno v:Algorithmica Ročník 82; číslo 5; s. 1298 - 1328
Hlavní autor: Damaschke, Peter
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.05.2020
Springer Nature B.V
Témata:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Approximation algorithm
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Fair division
Fragmentation
Linear-time algorithm
Mathematical analysis
Mathematics of Computing
Parameterized algorithm
Parameters
Theory of Computation
Weighted graph
68Q25
68W40
05C85
Fair division
Parameterized algorithm
68W25
Approximation algorithm
Weighted graph
05C05
68P10
Linear-time algorithm
ISSN:0178-4617, 1432-0541, 1432-0541
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Shrnutí:A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases F = 1 and F = 2 . For F = 1 , the max–min and min–max problems are solvable in linear time. The case F = 2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if m ≥ n - 1 , and a solution always exists in this case, in contrast to F = 1 . Moreover, the problem is fixed-parameter tractable in the parameter 2 m - n . (Note that this parameter measures the number of agents above the trivial threshold m = n / 2 .) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation F = 2 to this averaging problem, and some approximation results in cases when m is close to either n or n  / 2.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0178-4617
1432-0541
1432-0541
DOI:10.1007/s00453-019-00643-z