230,877 research outputs found
Anagram-free Graph Colouring
An anagram is a word of the form where is a non-empty word and
is a permutation of . We study anagram-free graph colouring and give bounds
on the chromatic number. Alon et al. (2002) asked whether anagram-free
chromatic number is bounded by a function of the maximum degree. We answer this
question in the negative by constructing graphs with maximum degree 3 and
unbounded anagram-free chromatic number. We also prove upper and lower bounds
on the anagram-free chromatic number of trees in terms of their radius and
pathwidth. Finally, we explore extensions to edge colouring and
-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with
'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov.
Minor changes based on referee feedbac
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
A Local Algorithm for the Sparse Spanning Graph Problem
Constructing a sparse spanning subgraph is a fundamental primitive in graph
theory. In this paper, we study this problem in the Centralized Local model,
where the goal is to decide whether an edge is part of the spanning subgraph by
examining only a small part of the input; yet, answers must be globally
consistent and independent of prior queries.
Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees,
cannot be constructed efficiently in this model. Therefore, we settle for a
spanning subgraph containing at most edges (where is the
number of vertices and is a given approximation/sparsity
parameter). We achieve query complexity of
, (-notation hides
polylogarithmic factors in ). where is the maximum degree of the
input graph. Our algorithm is the first to do so on arbitrary bounded degree
graphs. Moreover, we achieve the additional property that our algorithm outputs
a spanner, i.e., distances are approximately preserved. With high probability,
for each deleted edge there is a path of
hops in the output that connects its endpoints
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
Co-operation and Integration in Wood Energy Production
The aim of the study was to investigate the effects of co-operation and integration in large-scale wood energy production. The total procurement cost and yield of forest chips (small-sized trees and logging residues) delivered to the consumption plant were calculated for three harvesting strategies. In Alternative 1 individual stands were harvested. In Alternative 2 the harvesting of small-sized trees and logging residues was integrated within forest holdings. Alternative 3 included both co-operation between neighbouring forest holdings and the integration of harvesting. In integrated harvesting, small trees and logging residues were jointly chipped at intermediate storages.
The study material consisted of forest management planning information and forest maps, in digital form, for privately owned areas totaling 15000 ha, of which 3720 ha was forest. GIS data and costs models were used in constructing a production model for a power plant consuming 100000 m3 of forest chips per year.
Integration raised the harvestable small energy wood yield by 30.5% (Alternative 2) and 31.5% (Alternative 3). The corresponding values for all forest chips were 12.9% and 13.3%. The average cost of forest chips was 3.4% lower in Alternative 2 and 4.9% lower in Alternative 3 than in individual stand harvesting. The cost effects on the total production cost of small tree chips were greater than on the production cost of logging residues. Co-operation and integration broaden the raw-material base for wood energy and make the supply more even
Fast approximation of search trees on trees with centroid trees
Search trees on trees (STTs) generalize the fundamental binary search tree
(BST) data structure: in STTs the underlying search space is an arbitrary tree,
whereas in BSTs it is a path. An optimal BST of size can be computed for a
given distribution of queries in time [Knuth 1971] and centroid BSTs
provide a nearly-optimal alternative, computable in time [Mehlhorn
1977].
By contrast, optimal STTs are not known to be computable in polynomial time,
and the fastest constant-approximation algorithm runs in time
[Berendsohn, Kozma 2022]. Centroid trees can be defined for STTs analogously to
BSTs, and they have been used in a wide range of algorithmic applications. In
the unweighted case (i.e., for a uniform distribution of queries), a centroid
tree can be computed in time [Brodal et al. 2001; Della Giustina et al.
2019]. These algorithms, however, do not readily extend to the weighted case.
Moreover, no approximation guarantees were previously known for centroid trees
in either the unweighted or weighted cases.
In this paper we revisit centroid trees in a general, weighted setting, and
we settle both the algorithmic complexity of constructing them, and the quality
of their approximation. For constructing a weighted centroid tree, we give an
output-sensitive time algorithm, where is
the height of the resulting centroid tree. If the weights are of polynomial
complexity, the running time is . We show these bounds to be
optimal, in a general decision tree model of computation. For approximation, we
prove that the cost of a centroid tree is at most twice the optimum, and this
guarantee is best possible, both in the weighted and unweighted cases. We also
give tight, fine-grained bounds on the approximation-ratio for bounded-degree
trees and on the approximation-ratio of more general -centroid trees
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