scholarly journals Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus

2015 ◽  
Author(s):  
Radu Curticapean ◽  
Dániel Marx
Keyword(s):  
Lower Bounds ◽  
Perfect Matchings ◽  
Bounded Treewidth

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 On the Number of Planar Orientations with Prescribed Degrees

10.37236/801 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Stefan Felsner ◽  
Florian Zickfeld
Keyword(s):  
Lower Bounds ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Perfect Matchings ◽  
Asymptotic Enumeration ◽  
Combinatorial Structures ◽  
Schnyder Woods ◽  
Planar Map ◽  
Planar Maps

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with out-degrees prescribed by a function $\alpha:V\rightarrow {\Bbb N}$ unifies many different combinatorial structures, including the afore mentioned. We call these orientations $\alpha$-orientations. The main focus of this paper are bounds for the maximum number of $\alpha$-orientations that a planar map with $n$ vertices can have, for different instances of $\alpha$. We give examples of triangulations with $2.37^n$ Schnyder woods, 3-connected planar maps with $3.209^n$ Schnyder woods and inner triangulations with $2.91^n$ bipolar orientations. These lower bounds are accompanied by upper bounds of $3.56^n$, $8^n$ and $3.97^n$ respectively. We also show that for any planar map $M$ and any $\alpha$ the number of $\alpha$-orientations is bounded from above by $3.73^n$ and describe a family of maps which have at least $2.598^n$ $\alpha$-orientations.

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
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