scholarly journals Randomized on-line algorithms and lower bounds for computing large independent sets in disk graphs

2007 ◽  
Vol 155 (2) ◽  
pp. 119-136 ◽  
Author(s):  
Ioannis Caragiannis ◽  
Aleksei V. Fishkin ◽  
Christos Kaklamanis ◽  
Evi Papaioannou
Keyword(s):  
Lower Bounds ◽  
Independent Sets ◽  
On Line

Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing

2006 ◽  
Vol 36 (2) ◽  
pp. 354-393 ◽  
Author(s):  
Yair Bartal ◽  
Amos Fiat ◽  
Stefano Leonardi
Keyword(s):  
Lower Bounds ◽  
Line Graph ◽  
Graph Problems ◽  
On Line ◽  
Optical Routing

scholarly journals Two nonlinear lower bounds for on-line computations

1984 ◽  
Vol 60 (1-3) ◽  
pp. 1-11 ◽  
Author(s):  
Pavol Dūrī ◽  
Zvi Galil ◽  
Wolfgang Paul ◽  
Ruediger Reischuk
Keyword(s):  
Lower Bounds ◽  
On Line

LOWER BOUNDS FOR STREETS AND GENERALIZED STREETS

2001 ◽  
Vol 11 (04) ◽  
pp. 401-421 ◽  
Author(s):  
ALEJANDRO LÓPEZ-ORTIZ ◽  
SVEN SCHUIERER
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real Line ◽  
Upper Bound ◽  
Competitive Ratio ◽  
Search Strategy ◽  
The Real ◽  
Simple Polygons ◽  
On Line ◽  
Special Classes

We present lower bounds for on-line searching problems in two special classes of simple polygons called streets and generalized streets. In streets we assume that the location of the target is known to the robot in advance and prove a lower bound of [Formula: see text] on the competitive ratio of any deterministic search strategy—which can be shown to be tight. For generalized streets we show that if the location of the target is not known, then there is a class of orthogonal generalized streets for which the competitive ratio of any search strategy is at least [Formula: see text] in the L2-metric—again matching the competitive ratio of the best known algorithm. We also show that if the location of the target is known, then the competitive ratio for searching in generalized streets in the L1-metric is at least 9 which is tight as well. The former result is based on a lower bound on the average competitive ratio of searching on the real line if an upper bound of D to the target is given. We show that in this case the average competitive ratio is at least 9-O(1/ log D).

scholarly journals Linear syzygies of Stanley-Reisner ideals

2001 ◽  
Vol 89 (1) ◽  
pp. 117 ◽  
Author(s):  
V Reiner ◽  
V Welker
Keyword(s):  
Simplicial Complex ◽  
Lower Bounds ◽  
Monomial Ideal ◽  
Independent Sets ◽  
Free Resolution ◽  
Monomial Ideals ◽  
Minimal Free Resolution ◽  
Linear Syzygies ◽  
Alexander Dual ◽  
Syzygy Modules

We give an elementary description of the maps in the linear strand of the minimal free resolution of a square-free monomial ideal, that is, the Stanley-Reisner ideal associated to a simplicial complex $\Delta$. The description is in terms of the homology of the canonical Alexander dual complex $\Delta^*$. As applications we are able to prove for monomial ideals and $j=1$ a conjecture of J. Herzog giving lower bounds on the number of $i$-syzygies in the linear strand of $j^{th}$-syzygy modules show that the maps in the linear strand can be written using only $\pm 1$ coefficients if $\Delta^*$ is a pseudomanifold exhibit an example where multigraded maps in the linear strand cannot be written using only $\pm 1$ coefficients compute the entire resolution explicitly when $\Delta^*$ is the complex of independent sets of a matroid

scholarly journals Lower Bounds for On-line Graph Colorings

2014 ◽  
pp. 507-515 ◽  
Author(s):  
Grzegorz Gutowski ◽  
Jakub Kozik ◽  
Piotr Micek ◽  
Xuding Zhu
Keyword(s):  
Lower Bounds ◽  
Line Graph ◽  
Graph Colorings ◽  
On Line

scholarly journals Lower Bounds for Maximal Matchings and Maximal Independent Sets

2021 ◽  
Vol 68 (5) ◽  
pp. 1-30
Author(s):  
Alkida Balliu ◽  
Sebastian Brandt ◽  
Juho Hirvonen ◽  
Dennis Olivetti ◽  
Mikaël Rabie ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Independent Set ◽  
Independent Sets ◽  
Deterministic Algorithm ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Maximal Independent Set ◽  
Maximal Matching ◽  
Maximal Independent Sets ◽  
Open Question

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of  n .

Strong vb-dominating and vb-independent sets of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050002 ◽  
Author(s):  
Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Number Formula

Let [Formula: see text] be a graph. A vertex [Formula: see text] strongly (weakly) b-dominates block [Formula: see text] if [Formula: see text] ([Formula: see text]) for every vertex [Formula: see text] in the block [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in [Formula: see text] is strongly (weakly) b-dominated by some vertex in [Formula: see text]. The strong (weak) vb-domination number [Formula: see text] ([Formula: see text]) is the order of a minimum SVBD (WVBD) set of [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if [Formula: see text] is a vertex block independent set and for every vertex [Formula: see text] and every block [Formula: see text] incident on [Formula: see text], there exists a vertex [Formula: see text] in the block [Formula: see text] such that [Formula: see text] ([Formula: see text]). The strong (weak) vb-independence number [Formula: see text] ([Formula: see text]) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of [Formula: see text]. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.

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