# An Analytical Model for Interference Alignment in Broadcast Assisted VANETs
## Appendix A  
**Proof of Lemma 1. ** Constraint (26) with integer variable \(TU(t)\) is identical with the form listed in (28). By the definition of \(TU(t)\), if there exists any data transmission in time slot t, then \(TU(t)\) equals to 1, otherwise, it is 0. (1) If there exists any data transmission in time slot \(t\), then there must be one \(\delta\overset{i}{{}_{T}}(t)\) that not equals 0. Therefore, we have  
\[\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)\geq 1\]  
For the first inequality in (28), we have:  
\[TU(t)\leq 1-1/N+\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)/N<2\]  
where \(\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)/N\) is greater than or equal to 1. Therefore, we have \(TU(t)\) is less than or equal to \(\alpha\) which is greater than or equal to 1. For the second inequality in (28), we have:  
\[TU(t)\geq\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)/N=1/N>0\]  
Combining the above, we have:  
\[1/N\leq TU(t)<2\]  
Since the value of \(TU(t)\) can only be chosen from integer 0 and 1, we have \(TU(t)=1\), which is identical to the first equation in (28)  
(2) If there is no data transmission on time slot t, it is obvious that \(\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)\) equals 0. For the first inequality in (28), we have:  
\[TU(t)\leq 1-1/N+\sum\limits_{i=1:N}\delta\overset{i}{{}_{T}}(t)/N{=1-1/N<}1\]For the second inequality in (28), we have: \[TU(t)\geq\sum_{i=1:N}\delta\,^{i}_{\ T}(t)/N=0\] Combining the above, we have: \[0\leq TU(t)\leq 1-1/N<1\] Since the value of \(TU(t)\) can only be chosen from integer 0 and 1, we have \(TU(t)=0\), which is identical to the second equation in (28). Therefore, we draw the conclusion that (26) and (28) are identical to each other. **Proof of Lemma 2. ** Nonlinear constraint (16) combined with (17), (18) can be reformed in to the following linear constraint (29)-(31) During the communication, we have following cases:  
1. When vehicle \(i\) is in transmitting mode, we have \(\delta^{i}_{T}(t)=1\). In the same time slot, if vehicle \(j\) is in receiving mode, we have \(\delta^{j}_{R}(t)=1\). However, if session from vehicle \(i\) is NOT intended for \(j\), we have \(\lambda_{i,j}(t)=0\). In this case, we have \(il_{i,j}(t)=0\). 2. When vehicle \(i\) is in transmitting mode, we have \(\delta^{i}_{T}(t)=1\). In the same time slot, if vehicle \(j\) is in receiving mode, we have \(\delta^{j}_{R}(t)=1\). And if session from vehicle \(i\) is intended for \(j\), we have \(\lambda_{i,j}(t)=1\). In this case, we have \(il_{i,j}(t)=0\). 3. When vehicle \(i\) is in transmitting mode, we have \(\delta^{i}_{T}(t)=1\). In the same time slot, if vehicle \(j\) is NOT in receiving mode, we have \(\delta^{j}_{R}(t)=0\). In this case, we have \(\lambda_{i,j}(t)=0\). In this case, we have \(il_{i,j}(t)=0\). 4. When vehicle \(i\) is NOT in transmitting mode, we have \(\delta^{i}_{T}(t)=0\). In the same time slot, if vehicle \(j\) is in receiving mode, we have \(\delta^{j}_{R}(t)=1\). In this case, we have \(\lambda_{i,j}(t)=0\). In this case, we have \(il_{i,j}(t)=0\). 5. When vehicle \(i\) is NOT in transmitting mode, we have \(\delta^{i}_{T}(t)=0\). In the same time slot, if vehicle \(j\) is in receiving mode, we have \(\delta^{j}_{R}(t)=0\). In this case, we have \(\lambda_{i,j}(t)=0\). In this case, we have \(il_{i,j}(t)=0\). Combining the above five cases, we have the following Equation:  
\[il_{i,j}(t)=\delta\,^{i}_{\ T}(t)\cdot\delta\,^{j}_{\ R}\cdot(1-\lambda\,_{i,j}(t))= \delta\,^{i}_{\ T}(t)\cdot\delta\,^{j}_{\ R}-\delta\,^{i}_{\ T}(t)\cdot\delta \,^{j}_{\ R}\cdot\lambda\,_{i,j}(t)\]  
We have two items on the right-hand-side of the equation. One is in the quadratic form, the other one is in cubic form, which will lead to a relatively high complexity in solving the optimization problem. What is done next is to reshape the polynomial form of constraint into a linear one. Considering the following constraints with (10), (29)-(31):  
\[\left\{