scholarly journals Computing the Partition Function for Perfect Matchings in a Hypergraph

2011 ◽  
Vol 20 (6) ◽  
pp. 815-835 ◽  
Author(s):  
ALEXANDER BARVINOK ◽  
ALEX SAMORODNITSKY
Keyword(s):  
Partition Function ◽  
Polynomial Time ◽  
Pairwise Disjoint ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Perfect Matchings ◽  
Constant Factor ◽  
Higher Dimensional ◽  
Element Set

Given non-negative weightswSon thek-subsetsSof akm-element setV, we consider the sum of the productswS1⋅⋅⋅wSmover all partitionsV=S1∪ ⋅⋅⋅ ∪Sminto pairwise disjointk-subsetsSi. When the weightswSare positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial inmfactor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

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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