3,912 research outputs found
Faster Graph Coloring in Polynomial Space
We present a polynomial-space algorithm that computes the number independent
sets of any input graph in time for graphs with maximum degree 3
and in time for general graphs, where n is the number of
vertices. Together with the inclusion-exclusion approach of Bj\"orklund,
Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster
polynomial-space algorithm for the graph coloring problem with running time
. As a byproduct, we also obtain an exponential-space
time algorithm for counting independent sets. Our main algorithm
counts independent sets in graphs with maximum degree 3 and no vertex with
three neighbors of degree 3. This polynomial-space algorithm is analyzed using
the recently introduced Separate, Measure and Conquer approach [Gaspers &
Sorkin, ICALP 2015]. Using Wahlstr\"om's compound measure approach, this
improvement in running time for small degree graphs is then bootstrapped to
larger degrees, giving the improvement for general graphs. Combining both
approaches leads to some inflexibility in choosing vertices to branch on for
the small-degree cases, which we counter by structural graph properties
Coloring random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters and where the
proliferation of metastable states is responsible for the onset of complexity
in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR
A Coloring Algorithm for Disambiguating Graph and Map Drawings
Drawings of non-planar graphs always result in edge crossings. When there are
many edges crossing at small angles, it is often difficult to follow these
edges, because of the multiple visual paths resulted from the crossings that
slow down eye movements. In this paper we propose an algorithm that
disambiguates the edges with automatic selection of distinctive colors. Our
proposed algorithm computes a near optimal color assignment of a dual collision
graph, using a novel branch-and-bound procedure applied to a space
decomposition of the color gamut. We give examples demonstrating the
effectiveness of this approach in clarifying drawings of real world graphs and
maps
Coloring Artemis graphs
We consider the class A of graphs that contain no odd hole, no antihole, and
no ``prism'' (a graph consisting of two disjoint triangles with three disjoint
paths between them). We show that the coloring algorithm found by the second
and fourth author can be implemented in time O(n^2m) for any graph in A with n
vertices and m edges, thereby improving on the complexity proposed in the
original paper
Optimal Online Edge Coloring of Planar Graphs with Advice
Using the framework of advice complexity, we study the amount of knowledge
about the future that an online algorithm needs to color the edges of a graph
optimally, i.e., using as few colors as possible. For graphs of maximum degree
, it follows from Vizing's Theorem that bits of
advice suffice to achieve optimality, where is the number of edges. We show
that for graphs of bounded degeneracy (a class of graphs including e.g. trees
and planar graphs), only bits of advice are needed to compute an optimal
solution online, independently of how large is. On the other hand, we
show that bits of advice are necessary just to achieve a
competitive ratio better than that of the best deterministic online algorithm
without advice. Furthermore, we consider algorithms which use a fixed number of
advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to
this class of algorithms). We show that for bipartite graphs, any such
algorithm must use at least bits of advice to achieve
optimality.Comment: CIAC 201
- âŠ