scholarly journals Hitting minors on bounded treewidth graphs. III. Lower bounds

2020 ◽  
Vol 109 ◽  
pp. 56-77 ◽  
Author(s):  
Julien Baste ◽  
Ignasi Sau ◽  
Dimitrios M. Thilikos
Keyword(s):  
Lower Bounds ◽  
Bounded Treewidth

scholarly journals Maximum Induced Forests in Graphs of Bounded Treewidth

10.37236/3826 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Glenn G. Chappell ◽  
Michael J. Pelsmajer
Keyword(s):  
Lower Bounds ◽  
Nonnegative Integer ◽  
Maximum Degree ◽  
Open Problems ◽  
Maximum Order ◽  
Bounded Treewidth

Given a nonnegative integer $d$ and a graph $G$, let $f_d(G)$ be the maximum order of an induced forest in $G$ having maximum degree at most $d$. We seek lower bounds for $f_d(G)$ based on the order and treewidth of $G$.We show that, for all $k,d\ge 2$ and $n\ge 1$, if $G$ is a graph with order $n$ and treewidth at most $k$, then $f_d(G)\ge\lceil{(2dn+2)/(kd+d+1)}\rceil$, unless $G\in\{K_{1,1,3},K_{2,3}\}$ and $k=d=2$. We give examples that show that this bound is tight to within $1$.We conjecture a bound for $d=1$: $f_1(G) \ge\lceil{2n/(k+2)}\rceil$, which would also be tight to within $1$, and we prove it for $k\le 3$. For $k\ge 4$ the conjecture remains open, and we prove a weaker bound: $f_1(G)\ge (2n+2)/(2k+3)$. We also examine the cases $d=0$ and $k=0,1$.Lastly, we consider open problems relating to $f_d$ for graphs on a given surface, rather than graphs of bounded treewidth.

scholarly journals Parameterized Complexity of Envy-Free Resource Allocation in Social Networks

2020 ◽  
Vol 34 (05) ◽  
pp. 7135-7142
Author(s):  
Eduard Eiben ◽  
Robert Ganian ◽  
Thekla Hamm ◽  
Sebastian Ordyniak
Keyword(s):  
Social Networks ◽  
Resource Allocation ◽  
Social Network ◽  
Lower Bounds ◽  
Parameterized Complexity ◽  
Classical Problem ◽  
Full Information ◽  
Bounded Treewidth ◽  
Basic Model ◽  
The Social

We consider the classical problem of allocating resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and items. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has bounded treewidth or bounded clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved.

Two Lower Bounds for Shortest Double-Base Number System

2015 ◽  
Vol E98.A (6) ◽  
pp. 1310-1312 ◽  
Author(s):  
Parinya CHALERMSOOK ◽  
Hiroshi IMAI ◽  
Vorapong SUPPAKITPAISARN
Keyword(s):  
Lower Bounds ◽  
Number System ◽  
Base Number ◽  
Double Base

Lower bounds on the dimension of the Rauzy gasket

2020 ◽  
Vol 148 (2) ◽  
pp. 321-327
Author(s):  
Rodolfo Gutiérrez-Romo ◽  
Carlos Matheus
Keyword(s):  
Lower Bounds
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