307 research outputs found
On the Complexity of Digraph Colourings and Vertex Arboricity
It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular -colouring is
NP-complete for all rational . In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
Deciding if the star dichromatic number of a digraph is at most is
NP-complete for every rational .
Deciding if the fractional dichromatic number of a digraph is at most is
NP-complete for every .
Deciding if the circular vertex arboricity of a graph is at most is
NP-complete for every rational .
To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Shortest paths between shortest paths and independent sets
We study problems of reconfiguration of shortest paths in graphs. We prove
that the shortest reconfiguration sequence can be exponential in the size of
the graph and that it is NP-hard to compute the shortest reconfiguration
sequence even when we know that the sequence has polynomial length. Moreover,
we also study reconfiguration of independent sets in three different models and
analyze relationships between these models, observing that shortest path
reconfiguration is a special case of independent set reconfiguration in perfect
graphs, under any of the three models. Finally, we give polynomial results for
restricted classes of graphs (even-hole-free and -free graphs)
Hereditarily hard H-colouring problems
AbstractLet H be a graph (respectively digraph) whose vertices are called ‘colours’. An H-colouring of a graph (respectively digraph) G is an assignment of these colours to the vertices of G so that if u is adjacent to v in G, then the colour of u is adjacent to the colour of v in H. We continue the study of the complexity of the H-colouring problem ‘Does a given graph (respectively digraph) admit an H-colouring?’. For graphs it was proved that the H-colouring problem is NP-complete whenever H contains an odd cycle, and is polynomial for bipartite graphs. For directed graphs the situation is quite different, as the addition of an edge to H can result in the complexity of the H-colouring problem shifting from NP-complete to polynomial. In fact, there is not even a plausible conjecture as to what makes directed H-colouring problems difficult in general. Some order may perhaps be found for those digraphs H in which each vertex has positive in-degree and positive out-degree. In any event, there is at least, in this case, a conjecture of a classification by complexity of these directed H-colouring problems. Another way, which we propose here, to bring some order to the situation is to restrict our attention to those digraphs H which, like odd cycles in the case of graphs, are hereditarily hard, i.e., are such that the H′-colouring problem is NP-hard for any digraph H′ containing H as a subdigraph. After establishing some properties of the digraphs in this class, we make a conjecture as to precisely which digraphs are hereditarily hard. Surprisingly, this conjecture turns out to be equivalent to the one mentioned earlier. We describe several infinite families of hereditarily hard digraphs, and identify a family of digraphs which are minimal in the sense that it would be sufficient to verify the conjecture for members of that family
Graph homomorphisms with infinite targets
AbstractLet H be a fixed graph whose vertices are called colours. Informally, an H-colouring of a graph G is an assignment of these colours to the vertices of G such that adjacent vertices receive adjacent colours. We introduce a new tool for proving NP-completeness of H-colouring problems, which unifies all methods used previously. As an application we extend, to infinite graphs of bounded degree, the theorem of Hell and Nešetřil that classifies finite H-colouring problems by complexity
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs
We study Markov chains for -orientations of plane graphs, these are
orientations where the outdegree of each vertex is prescribed by the value of a
given function . The set of -orientations of a plane graph has
a natural distributive lattice structure. The moves of the up-down Markov chain
on this distributive lattice corresponds to reversals of directed facial cycles
in the -orientation. We have a positive and several negative results
regarding the mixing time of such Markov chains.
A 2-orientation of a plane quadrangulation is an orientation where every
inner vertex has outdegree 2. We show that there is a class of plane
quadrangulations such that the up-down Markov chain on the 2-orientations of
these quadrangulations is slowly mixing. On the other hand the chain is rapidly
mixing on 2-orientations of quadrangulations with maximum degree at most 4.
Regarding examples for slow mixing we also revisit the case of 3-orientations
of triangulations which has been studied before by Miracle et al.. Our examples
for slow mixing are simpler and have a smaller maximum degree, Finally we
present the first example of a function and a class of plane
triangulations of constant maximum degree such that the up-down Markov chain on
the -orientations of these graphs is slowly mixing
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