132 research outputs found
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
On Supergraphs Satisfying CMSO Properties
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence φ, a positive integer t, and a connected graph G of maximum degree at most Δ, and determines, in time f(|φ|,t)⋅2O(Δ⋅t)⋅|G|O(t), whether G has a supergraph G′ of treewidth at most t such that G′⊨φ. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)⋅2O(Δ⋅d)⋅|G|O(d), whether a connected graph of maximum degree Δ has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has an k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.publishedVersio
On Supergraphs Satisfying CMSO Properties
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence F, a positive integer t, and a connected graph G of maximum degree at most D, and determines, in time f(|F|,t)*2^O(D*t)*|G|^O(t), whether G has a supergraph G\u27 of treewidth at most t such that G\u27 satisfies F.
The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)*2^O(D*d)*|G|^O(d), whether a connected graph of maximum degree D has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has a k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d.
This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth
Making triangulations 4-connected using flips
We show that any combinatorial triangulation on n vertices can be transformed
into a 4-connected one using at most floor((3n - 9)/5) edge flips. We also give
an example of an infinite family of triangulations that requires this many
flips to be made 4-connected, showing that our bound is tight. In addition, for
n >= 19, we improve the upper bound on the number of flips required to
transform any 4-connected triangulation into the canonical triangulation (the
triangulation with two dominant vertices), matching the known lower bound of 2n
- 15. Our results imply a new upper bound on the diameter of the flip graph of
5.2n - 33.6, improving on the previous best known bound of 6n - 30.Comment: 22 pages, 8 figures. Accepted to CGTA special issue for CCCG 2011.
Conference version available at
http://2011.cccg.ca/PDFschedule/papers/paper34.pd
Algorithms for subnetwork mining in heterogeneous networks
International audienceSubnetwork mining is an essential issue in network analysis, with specific applications e.g. in biological networks, social networks, information networks and communication networks. Recent applications require the extraction of subnetworks (or patterns) involving several relations between the objects of interest, each such relation being given as a network. The complexity of a particular mining problem increases with the different nature of the networks, their number, their size, the topology of the requested pattern, the criteria to optimize. In this emerging field, our paper deals with two networks respectively represented as a directed acyclic graph and an undirected graph, on the same vertex set. The sought pattern is a longest path in the directed graph whose vertex set induces a connected subgraph in the undirected graph. This problem has immediate applications in biological networks, and predictable applications in social, information and communication networks. We study the complexity of the problem, thus identifying polynomial, NP-complete and APX-hard cases. In order to solve the difficult cases, we propose a heuristic and a branch-and-bound algorithm. We further perform experimental evaluation on both simulated and real data
Moving Vertices to Make Drawings Plane
A straight-line drawing of a planar graph need not be plane, but
can be made so by moving some of the vertices. Let shift denote the
minimum number of vertices that need to be moved to turn into a plane
drawing of . We show that shift is NP-hard to compute and to
approximate, and we give explicit bounds on shift when is a
tree or a general planar graph. Our hardness results extend to
1BendPointSetEmbeddability, a well-known graph-drawing problem.Comment: This paper has been merged with http://arxiv.org/abs/0709.017
Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications
Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable
A face cover perspective to embeddings of planar graphs
It was conjectured by Gupta et al. [Combinatorica04] that every planar graph
can be embedded into with constant distortion. However, given an
-vertex weighted planar graph, the best upper bound on the distortion is
only , by Rao [SoCG99]. In this paper we study the case where
there is a set of terminals, and the goal is to embed only the terminals
into with low distortion. In a seminal paper, Okamura and Seymour
[J.Comb.Theory81] showed that if all the terminals lie on a single face, they
can be embedded isometrically into . The more general case, where the
set of terminals can be covered by faces, was studied by Lee and
Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the
art is an upper bound of by Krauthgamer, Lee and Rika
[SODA19]. Our contribution is a further improvement on the upper bound to
. Since every planar graph has at most faces, any
further improvement on this result, will be a major breakthrough, directly
improving upon Rao's long standing upper bound. Moreover, it is well known that
the flow-cut gap equals to the distortion of the best embedding into .
Therefore, our result provides a polynomial time -approximation to the sparsest cut problem on planar graphs, for the
case where all the demand pairs can be covered by faces
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