1
A PPENDIX
Proof
3: The proof is based on the Chernoff Bound. For vi ∈ [0, 1] independent random variables,
Pof Theorem P
let S = i vi , ν = E[ i vi ], then
δ2 ν
P [S ≤ (1 − δ)ν] ≤ e− 2 .
P c2 c2
P c2 c2
P
c1 c1
1
2
2
2
2
Let Φ1i =
c2 ỹj wj . All Φi , Φi , Φj are sums of R.V’s ∈ [0, 1]. With
c2 ỹi wj(i),P , Φj =
c1 x̃i wi,j(i) , Φi =
E[Φ2i ] = âi , E[Φ2j ] = b̂j , we have the following
P [Φ2i ≥ (1 − δ)âi ] ≥ 1 − e−
δ 2 âi
2
, P [Φ2j ≥ (1 − δ)b̂j ] ≥ 1 − e−
δ 2 b̂j
2
.
Note that the aggregate marginal
(flow) in the first time slot is obtained from the flow in the rounded solution
P utility
c2
≥ (1 − δ)âi . Since E[Φ1i |Φ2i ] = a˜i = Φ2i /(1 − δ) ≥ âi , it can be shown that
in the second slot. Assume c2 ỹic2 wj(i),P
P [Φ1i ≥ (1 − δ)âi |Φ2i ] ≥ 1 − e−
δ 2 âi
2
.
Now to ensure a net flow of at least (1 − δ)âi at every PU and (1 − δ)b̂j at every SU with high probability, we need:
2
2 δ 2 âi
1
,
= 1− R
P [Φ1i , Φ2i ≥ (1 − δ)âi ] ≥ 1 − e− 2
K
2 2
δ 2 b̂
1
2
− 2j
P [Φj ≥ (1 − δ)b̂j ] ≥ 1 − e
= 1− R
.
K
q
q
R)
2 ln(K R )
. Then the approximation bound B is given by
=
This results in δ = 2 ln(K
âi
b̂j
P √
P q
q
âi + j b̂j
i
B = 1 − 2 ln (K R ) P
(1)
P
i âi +
j b̂j
P √
P
P
P √
P
âi + j b̂j
i âi +
j b̂j
i
P
P
is
maximized
when
â
=
b̂
=
â
=
. By the normalized weight assumption i âi +
i
j
2N
S
i âi +
j b̂j
c1
c2
P
P
P
maxi {wi,j(i)
,wj(i),P
}
R
R
≥ 1, then i âi + j b̂j ≥ Kc . Substituting into (1), we have the following
b̂
≤
K
.
If
c
=
c1
c
j
j
min {w
,w 2
}
i,j
i,j(i)
j(i),P
with high probability
r
B ≥1−
4cNS
ln (K R ).
KR