An exact combinatorial algorithm for minimum graph bisection

Abstract

We present a novel exact algorithm for the minimum graph bisection problem, whose goal is to partition a graph into two equally-sized cells while minimizing the number of edges between them. Our algorithm is based on the branch-and-bound framework and, unlike most previous approaches, it is fully combinatorial. We introduce novel lower bounds based on packing trees, as well as a new decomposition technique that contracts entire regions of the graph while preserving optimality guarantees. Our algorithm works particularly well on graphs with relatively small minimum bisections, solving to optimality several large real-world instances (with up to millions of vertices) for the first time.

Appendix: Algorithm examples

1.1 Packing bound

Figure 2 illustrates our lower bounds. Assume that $\epsilon = 0$ and that all vertices have unit weight. Note that $W=24$ and $W_-= W_+= 12$, i.e., we must find a solution with 12 vertices on each side. For simplicity, we refer to vertices by their rows and columns in the picture: vertex (1,1) is at the top left, and vertex (4,6) at the bottom right.

Assume some vertices have already been assigned to $A$ (red boxes) and $B$ (blue disks). First, we compute a flow $f$ (indicated in bold) from $A$ to $B$ (Fig. 2a). By removing these edges, we obtain the graph $G_f$. We then compute a tree packing on $G_f$, as shown in Fig. 2b. For each of the 11 edges $(u,v)$ with $u \in A$ and $v \not \in A$ we grow a tree (indicated by different colors and labeled $a, \dots , k$) in $G_f$. Note that the deadweight (number of vertices unreachable from $A$) is 6, and that 15 free vertices are reachable from $A$. Finally, we allocate the weights of these 15 vertices to the trees.

Figure 2c shows an allocation where each vertex is assigned in full to a single tree. This results in 7 trees of weight 1 ($b$, $c$, $e$, $f$, $i$, $j$, $k$), and 4 of weight 2 ($a$, $d$, $g$, $h$). Together, three of the heaviest trees have weight 6; with 6 units of deadweight, these trees are enough to reach the target weight of 12. Therefore, the packing bound is 3. Together with the flow bound, this gives a total lower bound of 6.

Figure 2d shows an alternative allocation in which some vertices are split equally among their incident trees. This results in 3 trees of weight 1 ($e$, $f$, $j$), and 8 trees of weight $1.5$ ($a$, $b$, $c$, $d$, $g$, $h$, $i$, $k$). Now, we must add at least 4 trees to $B$ to ensure its weight is at least 12. The packing bound is thus 4 and the total lower bound is 7, matching the optimum solution.

Fig. 2

Example for lower bounds. Red boxes and blue circles are already assigned to $A$ and $B$, respectively. The figures show a the maximum $A$-$B$ flow; b a set of maximal edge-disjoint trees rooted at $A$; c an integral vertex allocation; and d a fractional allocation where vertices with two labels have their weights equally split among the corresponding trees (color figure online)

1.2 Forced assignment

Figure 3 gives an example for our forced assignment techniques. We start from the tree packing in Fig. 2b. Figure 3a shows how the flow-based forced assignment applies to vertex $(2,2)$. It is incident to four trees ($b$, $e$, $f$, $g$). If it were assigned to $B$ (blue circles), the flow bound would increase by 4 units, to 7. Using the extended flow-based forced assignment, we can increase the flow bound by another 2 units, sending flow along $f+j$ and $b+a$ to $A$ (red squares). If the total lower bound, including the recomputed packing bound, is at least as high as the best solution seen so far, we can safely assign vertex $(2,2)$ to $A$.

Figure 3b illustrates our subdivision-based forced assignment. Consider what would happen if we were to assign vertex $(4,2)$ to $A$. We implicitly split all trees incident to this vertex ($i$ and $j$) into new trees $i'$, $i''$, and $j'$. Tree $i'$ is rooted at vertex $(4,3)$, and the others at vertex $(4,2)$. The vertex assignment remains consistent with the original one (as in Fig. 2c or 2(d), for example). We then recompute the packing bound for this set of trees. If the new lower bound is at least as high as the best solution seen so far, we can safely assign vertex $(4,2)$ to $B$.

Fig. 3

Examples of forced assignments. a Shows the additional flow that would be created if vertex $(2,2)$ were assigned to $B$ (blue circles); solid edges correspond to the standard flow-based forced assignment and dashed edges to the extended version. b Shows (with primed labels) new trees that would be created if vertex $(4,2)$ were assigned to $A$ (red squares) (color figure online)