A special role of Boolean quadratic polytopes among other combinatorial polytopes

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we us...

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Bibliographic Details
Published in:arXiv.org
Main Author: Maksimenko, Aleksandr
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 05.03.2015
Subjects:
Boolean algebra
Combinatorial analysis
Equivalence
Graph theory
Knapsack problem
Polytopes
Quadratic programming
ISSN:2331-8422
Online Access:Get full text
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Summary:We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family \(P\) is affinely reduced to a family \(Q\) if for every polytope \(p\in P\) there exists \(q\in Q\) such that \(p\) is affinely equivalent to \(q\) or to a face of \(q\), where \(\dim q = O((\dim p)^k)\) for some constant \(k\). Under this comparison the above-mentioned families are splitted into two equivalence classes. We show also that these two classes are simpler (in the above sence) than the families of poytopes of the following problems: set covering, traveling salesman, 0-1 knapsack problem, 3-satisfiability, cubic subgraph, partial ordering. In particular, Boolean quadratic polytopes appear as faces of polytopes in every of the mentioned families.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1408.0948