130 research outputs found
A new approach on locally checkable problems
By providing a new framework, we extend previous results on locally checkable
problems in bounded treewidth graphs. As a consequence, we show how to solve,
in polynomial time for bounded treewidth graphs, double Roman domination and
Grundy domination, among other problems for which no such algorithm was
previously known. Moreover, by proving that fixed powers of bounded degree and
bounded treewidth graphs are also bounded degree and bounded treewidth graphs,
we can enlarge the family of problems that can be solved in polynomial time for
these graph classes, including distance coloring problems and distance
domination problems (for bounded distances)
About equivalent interval colorings of weighted graphs
AbstractGiven a graph G=(V,E) with strictly positive integer weights ωi on the vertices i∈V, a k-interval coloring of G is a function I that assigns an interval I(i)⊆{1,…,k} of ωi consecutive integers (called colors) to each vertex i∈V. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0̸ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi=1 for all vertices i∈V).Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1,…,k such that ℓ∈I1(i) if and only if π(ℓ)∈I2(i) for all vertices i∈V. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions
Matrix norms and rapid mixing for spin systems
We give a systematic development of the application of matrix norms to rapid
mixing in spin systems. We show that rapid mixing of both random update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the
associated dependency matrix is less than 1. We give improved analysis for the
case in which the diagonal of the dependency matrix is (as in heat
bath dynamics). We apply the matrix norm methods to random update and
systematic scan Glauber dynamics for coloring various classes of graphs. We
give a general method for estimating a norm of a symmetric nonregular matrix.
This leads to improved mixing times for any class of graphs which is hereditary
and sufficiently sparse including several classes of degree-bounded graphs such
as nonregular graphs, trees, planar graphs and graphs with given tree-width and
genus.Comment: Published in at http://dx.doi.org/10.1214/08-AAP532 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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