Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
It remains to find the optimal sub-carrier pairing π. Substituting (2.29) into
(2.26), we obtain
D (ν, λ) = max
π
s.t.
K∑ K∑
M∑
π k,j F m ∗ ,(k,j)(ν, λ) + ν m P m + λQ (2.30)
k=1 j=1
K∑
π k,j = 1, ∀j,
k=1
j=1
m=1
K∑
π k,j = 1, ∀k. (2.31)
Let F be a K × K matrix such that
⎡
⎤
F m ∗ ,(1,1) F m ∗ ,(1,2) ... F m ∗ ,(1,K)
F m ∗ ,(2,1) F m ∗ ,(2,2) ... F m ∗ ,(2,K)
: : :
F =
. (2.32)
. . .
⎢F ⎣ m ∗ ,(K−1,1) F m ∗ ,(K−1,2) ... F m ∗ ,(K−1,K) ⎥
⎦
F m ∗ ,(K,1) F m ∗ ,(K,2) ... F m ∗ ,(K,K)
The matrix F can be considered as a profit matrix with row indices being different
customers and column indices being different items to be sold. The value of each
entry can be treated as the profit by selling a particular item to a particular customer.
Problem (2.30) is equivalent to maximizing the sum profit by choosing the best
selling strategy that can only sell one item to one customer. Such kind of linear
assignment problem can be solved efficiently from the standard Hungarian algorithm
with time complexity of O(K 3 ) [39]. The steps of Hungarian algorithm are briefly
described as follows [39]:
1) Subtract the values in each row from the maximum number in the row.
2) Subtract the minimum number in each column from the entire column.
3) Cover all zeroes in the matrix with as few lines (horizontal and/or vertical) as
possible.
4) If the number of lines equals the size of the matrix, find the solution. If the
number of lines is less than the size of the matrix, find the minimum number
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