126 research outputs found
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Parameterized complexity of Bandwidth of Caterpillars and Weighted Path Emulation
In this paper, we show that Bandwidth is hard for the complexity class
for all , even for caterpillars with hair length at most three.
As intermediate problem, we introduce the Weighted Path Emulation problem:
given a vertex-weighted path and integer , decide if there exists a
mapping of the vertices of to a path , such that adjacent vertices
are mapped to adjacent or equal vertices, and such that the total weight of the
image of a vertex from equals an integer . We show that {\sc Weighted
Path Emulation}, with as parameter, is hard for for all , and is strongly NP-complete. We also show that Directed Bandwidth is hard
for for all , for directed acyclic graphs whose underlying
undirected graph is a caterpillar.Comment: 31 pages; 9 figure
Linear orderings of random geometric graphs (extended abstract)
In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold
with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic
ordering for our layout problems on the class of random geometric graphs.Postprint (published version
Euclidean Networks with a Backbone and a Limit Theorem for Minimum Spanning Caterpillars
A caterpillar network (or graph) G is a tree with the property that removal of the leaf edges of Gleaves one with a path. Here we focus on minimum weight spanning caterpillars where the vertices are points in the Euclidean plane and the costs of the path edges and the leaf edges are multiples of their corresponding Euclidean lengths. The flexibility in choosing the weight for path edges versus the weight for leaf edges gives some useful flexibility in modeling. In particular, one can accommodate problems motivated by communications theory such as the “last mile problem.” Geometric and probabilistic inequalities are developed that lead to a limit theorem that is analogous to the well-known Beardwood, Halton, and Hammersley theorem for the length of the shortest tour through a random sample, but the minimal spanning caterpillars fall outside the scope of the theory of subadditive Euclidean functionals
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
On the Extended TSP Problem
We initiate the theoretical study of Ext-TSP, a problem that originates in
the area of profile-guided binary optimization. Given a graph with
positive edge weights , and a non-increasing discount
function such that and for , for some
parameter that is part of the problem definition. The problem is to
sequence the vertices so as to maximize , where is the position of
vertex~ in the sequence.
We show that \prob{Ext-TSP} is APX-hard to approximate in general and we give
a -approximation algorithm for general graphs and a PTAS for some sparse
graph classes such as planar or treewidth-bounded graphs.
Interestingly, the problem remains challenging even on very simple graph
classes; indeed, there is no exact time algorithm for trees unless
the ETH fails. We complement this negative result with an exact time
algorithm for trees.Comment: 17 page
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
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