1,997 research outputs found
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
Lower Bounds for the Graph Homomorphism Problem
The graph homomorphism problem (HOM) asks whether the vertices of a given
-vertex graph can be mapped to the vertices of a given -vertex graph
such that each edge of is mapped to an edge of . The problem
generalizes the graph coloring problem and at the same time can be viewed as a
special case of the -CSP problem. In this paper, we prove several lower
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound .
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound is almost asymptotically
tight.
We also investigate what properties of graphs and make it difficult
to solve HOM. An easy observation is that an upper
bound can be improved to where
is the minimum size of a vertex cover of . The second
lower bound shows that the upper bound is
asymptotically tight. As to the properties of the "right-hand side" graph ,
it is known that HOM can be solved in time and
where is the maximum degree of
and is the treewidth of . This gives
single-exponential algorithms for graphs of bounded maximum degree or bounded
treewidth. Since the chromatic number does not exceed
and , it is natural to ask whether similar
upper bounds with respect to can be obtained. We provide a negative
answer to this question by establishing a lower bound for any
function . We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page
Totally frustrated states in the chromatic theory of gain graphs
We generalize proper coloring of gain graphs to totally frustrated states,
where each vertex takes a value in a set of `qualities' or `spins' that is
permuted by the gain group. (An example is the Potts model.) The number of
totally frustrated states satisfies the usual deletion-contraction law but is
matroidal only for standard coloring, where the group action is trivial or
nearly regular. One can generalize chromatic polynomials by constructing spin
sets with repeated transitive components.Comment: 14 pages, 2 figure
Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements
Hyperplanes of the form x_j = x_i + c are called affinographic. For an
affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we
study the function f(M) that counts integral points in [1,M]^n that do not lie
in any hyperplane of the arrangement. We show that f(M) is a piecewise
polynomial function of positive integers M, composed of terms that appear
gradually as M increases. Our approach is to convert the problem to one of
counting integral proper colorations of a rooted integral gain graph. An
application is to interval coloring in which the interval of available colors
for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo
M; the number of proper modular colorations is a different piecewise polynomial
that for large M becomes the characteristic polynomial of the arrangement (by
which means Athanasiadis previously obtained that polynomial). We also study
this function for all positive moduli.Comment: 13 p
An elementary chromatic reduction for gain graphs and special hyperplane arrangements
A gain graph is a graph whose edges are labelled invertibly by "gains" from a
group. "Switching" is a transformation of gain graphs that generalizes
conjugation in a group. A "weak chromatic function" of gain graphs with gains
in a fixed group satisfies three laws: deletion-contraction for links with
neutral gain, invariance under switching, and nullity on graphs with a neutral
loop. The laws lead to the "weak chromatic group" of gain graphs, which is the
universal domain for weak chromatic functions. We find expressions, valid in
that group, for a gain graph in terms of minors without neutral-gain edges, or
with added complete neutral-gain subgraphs, that generalize the expression of
an ordinary chromatic polynomial in terms of monomials or falling factorials.
These expressions imply relations for chromatic functions of gain graphs.
We apply our relations to some special integral gain graphs including those
that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining
new evaluations of and new ways to calculate the zero-free chromatic polynomial
and the integral and modular chromatic functions of these gain graphs, hence
the characteristic polynomials and hypercubical lattice-point counting
functions of the arrangements. We also calculate the total chromatic polynomial
of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page
Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs
A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added
references, clarified examples. 35 p
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
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