scholarly journals Clustered Planarity Testing Revisited

10.37236/5002 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Radoslav Fulek ◽  
Jan Kynčl ◽  
Igor Malinović ◽  
Dömötör Pálvölgyi
Keyword(s):  
Polynomial Time ◽  
Planar Graphs ◽  
Classical Result ◽  
Short Proof ◽  
Matroid Intersection ◽  
Intersection Algorithm ◽  
Clustered Planarity ◽  
First Time ◽  
Straightforward Extension

The Hanani–Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani–Tutte theorem in the case when each cluster induces a connected subgraph.Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.

An Approximation Algorithm with Factor Two for a Repetitive Routing Problem of Grasp-and-Delivery Robots

2011 ◽  
Vol 15 (8) ◽  
pp. 1103-1108 ◽  
Author(s):  
Yoshiyuki Karuno ◽  
◽  
Hiroshi Nagamochi ◽  
Aleksandar Shurbevski ◽  
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Assembly Line ◽  
Routing Problem ◽  
Matroid Intersection ◽  
Current Configuration ◽  
Printed Circuit ◽  
Route Length ◽  
Intersection Algorithm ◽  
Electronic Parts

In this paper, we consider a routing problem for a single grasp-and-delivery robot used on a printed circuit (PC) board assembly line. The robot arranges n identical pins from their current configuration to the next required configuration by transferring them one by one in a transition. The n pins support a PC board from underneath to prevent it from overbending as an automated manipulator embeds electronic parts in the PC board from above. Each PC board has its own circuit pattern, and required configurations for PC boards all differ. Given an initial configuration of n pins and a sequence of m required configurations, the problem asks to find a transfer route of the robot that minimizes the route length over all m transitions. By applying a weighted matroid intersection algorithm, we show the repetitive routing problem to be 2-approximable in polynomial time.

scholarly journals Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

2010 ◽  
Vol 310 (21) ◽  
pp. 3049-3051 ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Short Proof

Separating Singularities of Holomorphic Functions

1998 ◽  
Vol 41 (4) ◽  
pp. 473-477 ◽  
Author(s):  
Jürgen Müller ◽  
Jochen Wengenroth
Keyword(s):  
Approximation Theory ◽  
Classical Result ◽  
Short Proof ◽  
Holomorphic Functions ◽  
Open Mapping ◽  
Complex Approximation ◽  
Basic Tool ◽  
Open Mapping Theorem ◽  
Separation Results

AbstractWe present a short proof for a classical result on separating singularities of holomorphic functions. The proof is based on the open mapping theorem and the fusion lemma of Roth, which is a basic tool in complex approximation theory. The same method yields similar separation results for other classes of functions.

scholarly journals Budget Feasible Mechanisms on Matroids

Algorithmica ◽  
2020 ◽  
Author(s):  
Stefano Leonardi ◽  
Gianpiero Monaco ◽  
Piotr Sankowski ◽  
Qiang Zhang
Keyword(s):  
Polynomial Time ◽  
Private Information ◽  
Independent Set ◽  
Deterministic Algorithm ◽  
Matroid Intersection ◽  
Practical Applications ◽  
Incentive Compatible ◽  
The Cost ◽  
Special Case ◽  
Budget Feasible

AbstractMotivated by many practical applications, in this paper we study budget feasible mechanisms with the goal of procuring an independent set of a matroid. More specifically, we are given a matroid $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) . Each element of the ground set E is controlled by a selfish agent and the cost of the element is private information of the agent itself. A budget limited buyer has additive valuations over the elements of E. The goal is to design an incentive compatible budget feasible mechanism which procures an independent set of the matroid of largest possible value. We also consider the more general case of the pair $${\mathcal {M}}=(E,{\mathcal {I}})$$ M = ( E , I ) satisfying only the hereditary property. This includes matroids as well as matroid intersection. We show that, given a polynomial time deterministic algorithm that returns an $$\alpha $$ α -approximation to the problem of finding a maximum-value independent set in $${\mathcal {M}}$$ M , there exists an individually rational, truthful and budget feasible mechanism which is $$(3\alpha +1)$$ ( 3 α + 1 ) -approximated and runs in polynomial time, thus yielding also a 4-approximation for the special case of matroids.

scholarly journals Graphs of systoles on hyperbolic surfaces

2019 ◽  
Vol 11 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Bidyut Sanki ◽  
Siddhartha Gadgil
Keyword(s):  
Planar Graphs ◽  
Classical Result ◽  
Hyperbolic Surface ◽  
Necessary Condition ◽  
Major Result ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Fat Graphs ◽  
Admissible Graph

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

Boundary Classes of Planar Graphs

2008 ◽  
Vol 17 (2) ◽  
pp. 287-295 ◽  
Author(s):  
VADIM LOZIN
Keyword(s):  
Polynomial Time ◽  
Planar Graphs ◽  
Dominating Set ◽  
Helpful Tool ◽  
Tree Width

We analyse classes of planar graphs with respect to various properties such as polynomial-time solvability of thedominating setproblem or boundedness of the tree-width. A helpful tool to address this question is the notion of boundary classes. The main result of the paper is that for many important properties there are exactly two boundary classes of planar graphs.

On fixed-point logic with counting

2000 ◽  
Vol 65 (2) ◽  
pp. 777-787 ◽  
Author(s):  
Jörg Flum ◽  
Martin Grohe
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Planar Graphs ◽  
Expressive Power ◽  
Descriptive Complexity ◽  
Main Motivation ◽  
First Order ◽  
Finite Structures ◽  
Tree Width

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.

SELF-STABILIZING ALGORITHMS FOR ORDERINGS AND COLORINGS

2005 ◽  
Vol 16 (01) ◽  
pp. 19-36 ◽  
Author(s):  
WAYNE GODDARD ◽  
STEPHEN T. HEDETNIEMI ◽  
DAVID P. JACOBS ◽  
PRADIP K. SRIMANI
Keyword(s):  
Distributed Algorithms ◽  
Polynomial Time ◽  
Graph Coloring ◽  
Planar Graphs ◽  
Special Cases ◽  
Legitimate State

A k-forward numbering of a graph is a labeling of the nodes with integers such that each node has less than k neighbors whose labels are equal or larger. Distributed algorithms that reach a legitimate state, starting from any illegitimate state, are called self-stabilizing. We obtain three self-stabilizing (s-s) algorithms for finding a k-forward numbering, provided one exists. One such algorithm also finds the k-height numbering of graph, generalizing s-s algorithms by Bruell et al. [4] and Antonoiu et al. [1] for finding the center of a tree. Another k-forward numbering algorithm runs in polynomial time. The motivation of k-forward numberings is to obtain new s-s graph coloring algorithms. We use a k-forward numbering algorithm to obtain an s-s algorithm that is more general than previous coloring algorithms in the literature, and which k-colors any graph having a k-forward numbering. Special cases of the algorithm 6-color planar graphs, thus generalizing an s-s algorithm by Ghosh and Karaata [13], as well as 2-color trees and 3-color series-parallel graphs. We discuss how our s-s algorithms can be extended to the synchronous model.

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