3 research outputs found
Obstructions to Faster Diameter Computation: Asteroidal Sets
Full version of an IPEC'22 paperAn extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every -edge graph in can be computed in deterministic time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive -approximation of all vertex eccentricities in deterministic time. This is in sharp contrast with general -edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in time for any . As important special cases of our main result, we derive an -time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an -time algorithm for this problem on graphs of asteroidal number at most . We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions
Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates
We introduce notions of certificates allowing to bound eccentricities in a
graph. In particular , we revisit radius (minimum eccentricity) and diameter
(maximum eccentricity) computation and explain the efficiency of practical
radius and diameter algorithms by the existence of small certificates for
radius and diameter plus few additional properties. We show how such
computation is related to covering a graph with certain balls or complementary
of balls. We introduce several new algorithmic techniques related to
eccentricity computation and propose algorithms for radius, diameter and all
eccentricities with theoretical guarantees with respect to certain graph
parameters. This is complemented by experimental results on various real-world
graphs showing that these parameters appear to be low in practice. We also
obtain refined results in the case where the input graph has low doubling
dimension, has low hyperbolicity, or is chordal
Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates
We introduce notions of certificates allowing to bound eccentricities in a graph. In particular , we revisit radius (minimum eccentricity) and diameter (maximum eccentricity) computation and explain the efficiency of practical radius and diameter algorithms by the existence of small certificates for radius and diameter plus few additional properties. We show how such computation is related to covering a graph with certain balls or complementary of balls. We introduce several new algorithmic techniques related to eccentricity computation and propose algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various real-world graphs showing that these parameters appear to be low in practice. We also obtain refined results in the case where the input graph has low doubling dimension, has low hyperbolicity, or is chordal