Why the Goemans-Williamson's MAX-CUT algorithm relax the variables to vectors of $n-$dimension on unit sphere?

Question

Why not to some constant like 3 or 4 dimension? I suspect that it is because Cholesky Decompostion will work only for $n \times n$ matrix $B$ where $B^TB = P$ where $P$ is a semidefinite matrix. Is it true? Or there is something else?

Answer 1

The MAX-CUT algorithm relies on semidefinite programming, a convex optimization problem which can be solved in polynomial time. What you solve directly is the semidefinite program

$$ \begin{align*} & \max \sum_{ij} w_{ij} \frac{1-a_{ij}}{2} \\ s.t. \quad& (a_{ij})_{i,j=1}^n \text{ is positive semidefinite} \\ & a_{ii} = 1 \text{ for all $i$} \end{align*} $$

Given a solution for this program, you can extract the vectors using Cholesky decomposition.

The problem you are interested in is different:

$$ \begin{align*} & \max \sum_{ij} w_{ij} \frac{1-\langle v_i, v_j \rangle}{2} \\ s.t. \quad& v_i \in \mathbb{R}^d \\ & \|v_i\|^2 = 1 \end{align*} $$

When $d = 1$, this is just MAX-CUT. When $d$ is constant, optimizing over $\{v_i \in \mathbb{R}^d : \|v_i\| = 1 \}$ is likely NP-hard. When $d = n$, this can be solved using the semidefinite program described above. This is why we allow the dimension to be unbounded.

If you could force the dimension to be small, then you could improve the approximation ratio of the algorithm, see for example Rounding Two and Three Dimensional Solutions of the SDP Relaxation of MAX CUT by Avidor and Zwick. Assuming the unique games conjecture, however, the approximation guarantee of the Goemans–Williamson algorithm is tight. This shows that even in the particular case of MAX-CUT, it is likely that finding a low-dimensional solution cannot be done efficiently.