Extending the Support for Separation

Helmberg observed in [71] that the quality of the SDP relaxation may be improved considerably if the support ¯E, on which we call the separation routines, includes node pairs_{{i, j} /∈ E. This} is due to the fact that any Xij ∈ {−1, 1} with i 6= j describes the relation between nodes i and j

regardless whether{i, j} ∈ E or not. Therefore, valid inequalities can be defined on all variables yij and Xij and not just on those corresponding to edges{i, j} ∈ E. Moreover, all entries of the

relaxed variable X∈ [−1, 1]|V |×|V |_{will contain meaningful values, since they are all related via the}

variables on edges that are not in E can force some variables of edges in E to change.

One can also think of (theoretically) completing the graph G by adding the missing edges with edge costs zero. The minimum bisection problem would have the same optimal solution, but all edges would be available for separation. However, this full completion of G is not a practical approach for three reasons. First, the run times of our separation procedures depend on the number of edges in ¯E. Second, the primal approximation ˜X of (4.4) will only contain reliable entries ˜Xij if ¯Wk of (4.3) is aggregated on entry ij. Aggregating on all ij is inefficient for large

graphs, with say more than 1000 nodes, both from points of view of memory consumption and computation times. Third, the eigenvalue computations of the spectral bundle method will become very expensive if the aggregated support of all inequalities in the SDP relaxation becomes dense and the corresponding coefficient matrices do not have some special structure like that of the bisection constraint. Therefore, a reasonable selection of ¯E has to be made.

One can imagine two types of support extensions, those independent of the inequalities to be separated and those tailored for the inequalities. Extensions of the first type would be applied only once at the beginning of the computations of a relaxation, while extensions of the second type would be used on a regular basis during the computations. Below, we will explain the support extensions that we tested with the odd cycle inequalities and the bisection knapsack walk inequalities. Support Extension One-star (OS)

The inclusion of the star for the polyhedral bisection constraint of (4.2) constitutes a support extension of the first type which will always be present in ¯E. In particular, this support extension increases the support for triangle inequalities, because any edge in ¯E connecting two nodes of the star different from the centre node gives rise to a triangle via the centre node. There are | ¯E_{| − (|V | − 1) of these triangles.}

Support Extension λ2-max (λ2-max)

We already explained that all edges{i, j}, regardless whether they are in E or not, are linked via the bisection constraint and in particular via the positive semidefiniteness constraint. If we force the entries of X on some edges to attain certain values, e.g., by primal cutting planes, the solution X will have to adapt eventually to remain positive semidefinite. It can do so by altering some entries that are not yet restricted by cutting planes. The idea of the following support extension is to spread the support in some sense evenly so that the solution X will eventually be forced to cut original edges. Then, the primal objective value of the relaxation will increase and the dual may follow.

To be more precise, we introduce the so-called expansion constant of a graph. Definition 130 ([92]). The expansion constant of a graph G = (V, E) is

h(G) = min

S⊆V,|S|≤|V |2

|δ(S)| |S| .

A high expansion constant means that there is a guaranteed minimal number of edges in the cut defined by any arbitrarily chosen subset S _{⊆ V . (Families of) graphs with a guaranteed minimal}

expansion constant are also called expander graphs. Following the idea explained above, we would like the graph induced by ¯E to have a large expansion constant. The following result can be shown. Proposition 131 ([92]). Let G = (V, E) be a finite, connected, k-regular graph without loops. Let λ2 be the second smallest eigenvalue of the Laplace matrix L of G. Then

λ2

2 ≤ h(G) .

We have already introduced the second smallest eigenvalue λ2 of the Laplace matrix in Sec-

tion 1.4 in the context of spectral solution approaches to graph partitioning problems. We know that a large value of λ2, which is also called the algebraic connectivity of G, means that G is highly

connected. It is not too surprising that a highly connected graph also exhibits a high expansion constant. Thus, we now try to maximise λ2 of G[ ¯E] by inserting a given number k of additional

edges into ¯E. This problem is nontrivial and we use a heuristic proposed by Boyd and Ghosh [29]: Given the current support ¯E, we choose the next edge _{{i, j} /∈ ¯}E to be added to ¯E as one that maximises (vi− vj)2, where v is a unit eigenvector to λ2 L GE

.¯

Support Extension Random-degree-k (RD-k)

This heuristic is based on the observation that random regular graphs have a large expansion constant with a high probability (see, e.g., [120]). Given a parameter k _{∈ N, we add each edge} {i, j}, i < j, to ¯E with probability _{|V |}k .

Support Extension Shortest-path (SP)

A support extension heuristic tailored to the odd cycle inequalities was proposed by Helmberg in [71]. For each root node ¯v ∈ V , a shortest path tree with respect to edge weights 1 −

    ˜ Xij √_˜ XiiX˜jj     is computed on the graph induced by the current support ¯E. Each edge{¯v, ˜v} /∈ ¯E induces a cycle Cv˜¯v with respect to the shortest path tree. For each cycle Cv˜¯v, an odd set F¯v˜v is determined so

that the odd cycle inequality on Cv˜¯v and Fv˜¯v is maximally violated with respect to ˜X among all

odd cycle inequalities on Cv˜¯v. For each root node ¯v, the edge{¯v, ˜v} that creates the most violated

odd cycle using C¯v˜v and F¯v˜v is added to ¯E. The hope is that these newly created cycles will be

the support of violated odd cycle inequalities in the next separation round. In addition, for each ¯

v, the edge minimising X˜¯vv

 with {¯v, v} /∈ ¯E is added to ¯E, because the relation of nodes ¯v and v seems to be not decided yet.

Support Extension Most-decided (MD)

This heuristic looks at the needs of knapsack tree inequalities and bisection knapsack walk inequal- ities. Observe that the knapsack tree inequality (2.25) requires a tree support on a large number of nodes but with edge values ye close to zero, i.e., Xe close to 1. The support of a violated

knapsack tree inequality will never contain a path with two incident edges{v1, v2} and {v2, v3} so

that yv1v2+ yv2v3 > 1. In particular, we will encounter such a situation if for some reason, e.g.,

because of branching, yv1v2 = 1 and yv2v3 = 1. In terms of X, this means that Xv1v2 =−1 and

the positive semidefiniteness of the principal submatrix    Xv1v1 Xv1v2 Xv1v3 Xv2v1 Xv2v2 Xv2v3 Xv3v1 Xv3v2 Xv3v3    .

However, Xv1v3 = 1 is equivalent to yv1v3= 0. Thus, the edge{v1, v3} would be an ideal candidate

for inclusion into the tree support of the knapsack tree inequality. Therefore, we should insert {v1, v3} into the support ¯E. Moreover, observe that the support extension Shortest-path, as

described above, would (on the basis of the triangle {v1, v2, v3}) not consider edge {v1, v3} for

inclusion, because there is no violated triangle inequality for edge values yv1v2 = yv2v3 = 1 and

yv1v3 = 0.

Let us now turn to the bisection knapsack walk inequalities. They prefer edges e with values ye close to 0 or 1. In terms of Xe, these are values close to 1 or −1, respectively. Therefore, we

propose to extend the support in the following way. Let the nodes of V be sorted v1, . . . , v_{|V |}

correspondingly to the order of their columns in X. For each node vi, extend the support ¯E by an

edge{vi, vj} with Xij = min{Xij :{vi, vj} /∈ ¯E, i < j}, and an edge {vi, vj} with Xij = max{Xij :

{vi, vj} /∈ ¯E, i < j}.

Support Extension Try-knapsack-star (TKS)

This support extension is designed for the even bisection knapsack walk inequalities and is based on Proposition 60. Suppose, we have a star (VS, ES) centred at some node r in the subgraph of

G induced by ¯E. If all odd cycle inequalities on the subgraph induced by VS are fulfilled, then we

know from Proposition 60 that the strongest even bisection knapsack walk inequality rooted at r is the respective knapsack star inequality. The idea of the heuristic is to check for every centre node r_{∈ V the corresponding knapsack star inequality on the complete graph, and to extend the} support ¯E by the edges of a star that gives rise to the most violated inequality with respect to the unscaled ˜X.