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On page 338 of Williamson & Shmoys's The Design of Approximation Algorithms, the presentation of the ARV algorithm for the sparsest cut over a graph $G(V,E)$ has the following formulation

minimize $\frac{1}{n^2} \sum_{e=(i,j) \in E} c_e \lVert{v_i-v_j}\rVert^2$

subject to $\sum_{i,j \in V: i \neq j } \lVert{v_i-v_j}\rVert^2 = n^2 \\ \lVert{v_i-v_j}\rVert^2 \leq \lVert{v_i-v_k}\rVert^2 + \lVert{v_k-v_j}\rVert^2 \quad \forall i,j,k \in V \\ v_i \in R^n \quad \forall i \in V$

where $c_e$ is the weight of each edge. The authors state that this is an SDP relaxation of the original problem. Why is this the case?

To be an SDP relaxation the objective must be expressed as $tr(C^TX)=\sum_{i=1}^n\sum_{j=1}^nC_{ij}X_{ij} $ where $X$ is symmetric and positive semidefinite (that is $X \ge 0)$. In my attempts matrix $C$ is defined as

$C=[C_{ij}] \quad i=1..n,j=1..n$ where $C=\begin{cases} c_e & \text{if } e=(i,j) \in E \\ 0 & \text{otherwise} \end{cases}$.

My concern is with the definition of $X$. My first attempt was to define it as $$ X = [X_{ij}] = [\lVert{v_i-v_j}\rVert^2], $$ but this matrix is not positive semidefinite (it can be shown with a counterexample on $z^TXz \ge 0$ with $z_1z_2 \lt 0$ and $z_i=0$ otherwise).

My second and more promising attempt was to work a bit on the objective using that, $$ \lVert{v_i-v_j}\rVert^2 = \lVert{v_i}\rVert^2 + \lVert{v_j}\rVert^2 -2\langle v_i,v_j \rangle, $$ and define $X$ as $$ X = [X_{ij}] = [ \langle v_i,v_j \rangle ]. $$ This could work if I added the constraint that $\lVert{v_i}\rVert=1 ,\quad \forall i \in V$. In that case the objective would be a linear combination of inner products and $X$ would have a Cholesky decomposition of $X=V^TV \text{ with } V=[v_1...v_n]$.

But the authors explicitly state "the vector $v_i$ are not constrained to be unit vectors". Am I missing something here, should I somehow use the first constraint $\sum_{i,j \in V: i \neq j } \lVert{v_i-v_j}\rVert^2 = n^2$?

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You were on the right track with the parametrization $\lVert{v_i-v_j}\rVert^2 = \lVert{v_i}\rVert^2 + \lVert{v_j}\rVert^2 -2\langle v_i,v_j \rangle$ you proposed. Note that using your defintion $X = [X_{ij}] = [ \langle v_i,v_j \rangle ]$, each term $\lVert{v_i-v_j}\rVert^2$ becomes a linear function of $X$: $\lVert{v_i-v_j}\rVert^2 = X(i,i) + X(j,j) - 2 X(i,j)$, so the objective is linear with respect to $X$.

To express all the squared distances in matrix form , define the distance matrix $D\in R^{N\times N}, D(i,j) = \lVert{v_i-v_j}\rVert^2$. Then you get $D = e\,diag(X)^T + diag(X)\,e^T - 2X$, where $e$ is the vector of all ones, and your objective function is $Tr(C^T\,D) = Tr(C^T\,e\,diag(X)^T) + Tr(C^T\,diag(X)\,e^T) - 2 Tr(C^T\,X)$.

Using the cyclic property of the trace, you have that $Tr(C^T\,e\,diag(X)^T) = Tr(diag(X)^T C^T\,e\,)$. Then we can use the property that for (suitable sized) matrices $A$ and $B$ and a vector $c$, $Tr(c^T A \,diag(B)) = Tr(Diag(c^T A)\, B)$, where $diag(X)\in R^N$ is the diagonal of $X$ and $Diag(c)\in R^{N\times N}$ is a diagonal matrix whose diagonal is $c$. Putting it all together, the objective can be reformulated as:

$Tr(C^T\,D) = Tr(Diag(C^T\,e)\,X) + Tr(Diag(C\,e)\,X) - 2 Tr(C^T\,X) = Tr(\hat{C}^T X)$, for $\hat{C} = Diag(C^T\,e) + Diag(C\,e) - 2 C$

and you get the objective of a standard SDP, and using the same tools, you can also express the constraints as linear (in)equalities w.r.t. $X$.

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