Colouring Non-sparse Random Intersection Graphs

An intersection graph of

n

vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random Intersection Graphs (RIGs) (as defined in [18,32]) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (

m

in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability

p

. Such graphs nicely capture interactions in networks due to sharing of resources among nodes. We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs. We concentrate in a range of parameters not examined in the literature, namely: (a)

m

 = 

n

α

for

α

less than 1 (in this range, RIGs differ substantially from the Erdös-Renyi random graphs) and (b) the selection probability

p

is quite high (e.g. at least

$\frac{\ln^2{n}}{m}$

in our algorithm) and disallows direct greedy colouring methods.

We manage to get the following results:

For the case

mp

 ≤ 

β

ln

n

, for any constant

β

 < 1 − 

α

, we prove that

np

colours are enough to colour most of the vertices of the graph with high probability (whp). This means that even for quite dense graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph. Note also that this range of values of

m

,

p

is quite wider than the one studied in [4].

We propose and analyze an algorithm CliqueColour for finding a proper colouring of a random instance of

${\cal G}_{n, m, p}$

, for any

mp

 ≥ ln

2

n

. The algorithm uses information of the label sets assigned to the vertices of

G

n

,

m

,

p

and runs in

$O\left(\frac{n^2mp^2}{\ln{n}} \right)$

time, which is polynomial in

n

and

m

. We also show by a reduction to the uniform random intersection graphs model that the number of colours required by the algorithm are of the correct order of magnitude with the actual chromatic number of

G

n

,

m

,

p

.

We finally compare the problem of finding a proper colouring for

G

n

,

m

,

p

to that of colouring hypergraphs so that no edge is monochromatic. We show how one can find in polynomial time a

k

-colouring of the vertices of

G

n

,

m

,

p

, for any integer

k

, such that no clique induced by only one label in

G

n

,

m

,

p

is monochromatic.

Our techniques are novel and try to exploit as much as possible the hidden structure of random intersection graphs in this interesting range.