scholarly journals A polynomial-time algorithm for a stable matching problem with linear valuations and bounded side payments

2008 ◽  
Vol 25 (1) ◽  
pp. 83-98 ◽  
Author(s):  
Rashid Farooq
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Stable Matching ◽  
Matching Problem ◽  
Side Payments

COMPUTATIONAL COMPLEXITY OF THE PERFECT MATCHING PROBLEM IN HYPERGRAPHS WITH SUBCRITICAL DENSITY

2010 ◽  
Vol 21 (06) ◽  
pp. 905-924 ◽  
Author(s):  
MAREK KARPIŃSKI ◽  
ANDRZEJ RUCIŃSKI ◽  
EDYTA SZYMAŃSKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Existence Problem ◽  
Matching Problem ◽  
Uniform Hypergraphs ◽  
The Status ◽  
Np Complete

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for [Formula: see text] and trivial for c ≥ ½, leaving the status of the problem with c in the interval [Formula: see text] widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

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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