96 research outputs found
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
Parallel O(log(n)) time edge-colouring of trees and Halin graphs
We present parallel O(log(n))-time algorithms for optimal edge colouring of trees and Halin graphs with n processors on a a parallel random access machine without write conflicts (P-RAM). In the case of Halin graphs with a maximum degree of three, the colouring algorithm automatically finds every Hamiltonian cycle of the graph
Edge-colouring graphs with local list sizes
The famous List Colouring Conjecture from the 1970s states that for every
graph the chromatic index of is equal to its list chromatic index. In
1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds
asymptotically. Our main result is a local generalization of Kahn's theorem.
More precisely, we show that, for a graph with sufficiently large maximum
degree and minimum degree , the following
holds: for every assignment of lists of colours to the edges of , such that
for
each edge , there is an -edge-colouring of . Furthermore, Kahn
showed that the List Colouring Conjecture holds asymptotically for linear,
-uniform hypergraphs, and recently Molloy generalized Kahn's original result
to correspondence colouring as well as its hypergraph generalization. We prove
local versions of all of these generalizations by showing a weighted version
that simultaneously implies all of our results.Comment: 22 page
Transversal polynomial of r-fold covers
We construct a cover of a graph by blowing up each vertex to a set of
vertices and joining each pair of sets corresponding to adjacent vertices by a
matching with edges. To each cover of we associate a polynomial
, called the transversal polynomial. The coefficient of
is the number of -edge induced subgraphs of whose vertex set
is a transversal of the set system given by the blown-up vertices. We show that
satisfies a contraction-deletion formula, and that if and
the cover has index , then . We see that
has interesting connections to unique label covers and
correspondence colouring.Comment: 11 pages, 2 figure
Optimally edge-colouring outerplanar graphs is in NC
We prove that every outerplanar graph can be optimally edge-coloured in polylogarithmic time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM)
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
- …