NDLI: Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

Content Provider	Springer Nature Link
Author	Boros, Endre Elbassioni, Khaled Gurvich, Vladimir Khachiyan, Leonid
Copyright Year	2003
Abstract	A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most δ p +1, where δ is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors ℬ in the product of n lattices ℒ=ℒ1×⋯×ℒ n , where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into ℬ. We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors xℒ=ℒ1×⋯×ℒ n for a system of polymatroid inequalities ${{f_1(x) \ge t_1,\ldots,f_r(x) \ge t_r}}$ does not exceed max{Q,βlog t/c(2Q,β) }, where β is the number of minimal feasible vectors for the system, ${{Q=\|{{\mathcal L}}_1\|+\ldots+\|{{\mathcal L}}_n\|}}$ , ${{t=\hbox{max}\{t_1,\ldots,t_r\}}}$ , and c(ρ,β) is the unique positive root of the equation 2 c (ρ c/ logβ−1)=1. This bound is nearly sharp for the Boolean case ℒ={0,1} n , and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides ${{t_1, \ldots, t_r}}$ .
Starting Page	355
Ending Page	368
Page Count	14
File Format	PDF
ISSN	00255610
Journal	Mathematical Programming
Volume Number	98
Issue Number	1-3
e-ISSN	14364646
Language	English
Publisher	Springer-Verlag
Publisher Date	2003-04-10
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	dualization hypergraph incremental algorithm maximal independent set lattice polymatroid function system of polymatroid inequalities proper mapping
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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