Hypergraph Modeling and Approximation Algorithms for the Minimum Length Link Scheduling in Multiuser MIMO Networks

Algorithm 2

Algorithm 𝐃-𝐆(𝐆_𝐇): Providing a coloring solution for hypergraph 𝐺_𝐻=(𝑉_𝐻,𝐸_𝐻)

Input: An 𝑟-uniform hypergraph 𝐺_𝐻=(𝑉_𝐻,𝐸_𝐻).

Output: A coloring of 𝐺_𝐻.

(1)  𝑖=1; 𝑊=𝑉_𝐻;

(2)  while  𝑊≠∅

(3)  QuickSort (𝐺^𝑊_𝐻);   % sort 𝐺^𝑊_𝐻 by the DOI item in a decreasing order %

(4)  𝑈=∅

(5)  while  (𝑗≤|𝑊|)

(6)    if  𝑈∪{𝑊(𝑗)} is an independent set

(7)     𝑈=𝑈∪{𝑊(𝑗)}; % 𝑊(𝑗) is the 𝑗th element of 𝑊 %

(8)    else

(9)     break;

(10)    endif

(11)   endwhile

(12)  𝐶_𝑖=𝑈;  % color 𝑈 by color 𝑖 %

(13)  𝑖=𝑖+1;

(14)  𝑊=𝑊⧵𝑈;

(15) endwhile;

(16) return a coloring 𝐶={𝐶₁,𝐶₂,…,𝐶_𝑖−1};