Minimum Bisection

Abisection (B, W) in a graphG= (V, E) is a partition of its vertex set into two setsBandW, called the black and the white set, of sizes differing by at most one. An edge{x, y}ofGiscut by the bisection (B, W) ifx∈B andy∈W or vice versa. The number of edges cut by the bisection (B, W) is called the width of the bisection and is denoted byeG(B, W). Aminimum bisection in the graphGis a bisection (B^∗, W^∗) with

eG(B^∗, W^∗) = min{eG(B, W): (B, W) is a bisection inG}=: MinBis(G),

see Figure 1.1 for an example. Determining a bisection of minimum width is a famous optimization problem that is known to be NP-hard since 1976, see Theorem 1.3 in [GJS76]. In the following, we denote this problem byMinimum Bisection.

One way to deal with the hardness of a problem is to restrict the class of considered graphs. For example, when restricting theMinimum Bisection Problem to trees, it becomes solvable in polynomial time.

Indeed, there is an algorithm that computes inO(n³) time a minimum bisection in a tree onnvertices, see Theorem 4.3 in [Jan+05]. This algorithm by Jansen et al. relies on dynamic programming and can also be applied totree-likegraphs, i. e., graphs of constant tree-width. Roughly speaking, thetree-width of a graphGmeasures how tree-likeGis. For example, trees have tree-width 1, cycles and cacti, i. e., graphs

a)A bisection (B, W) inGwitheG(B, W) = 5. b) A minimum bisection inG.

Figure 1.1: A graphGand two bisections (B, W). In both parts, the vertices inB are colored black and the vertices inW are colored white. Each edge ofGthat is cut by (B, W) is colored red.

where each edge is contained in at most one cycle, have tree-width 2, and a complete graph onnvertices has tree-widthn−1. Returning to the algorithm in [Jan+05], when a tree decomposition of widthtof a graph onnvertices is provided as input, then the algorithm computes a minimum bisection in O(2^tn³) time.

Using the algorithm in [Bod96] to compute a tree decomposition, theMinimum Bisection Problem becomes polynomially tractable for graphs of constant tree-width. As there are planar graphs onnvertices that do not allow a tree decomposition of width less than√

n, the algorithm presented in [Jan+05] to compute a minimum bisection does not run in polynomial time for all planar graphs. In fact, it is open whether the Minimum Bisection Problemremains NP-hard when restricted to planar graphs. Díaz and Mertzios [DM14] believe that this is the case, since planar graphs and unit disk graphs often behave similarly with respect to computational complexity of optimization problems and they showed that the Minimum Bisection Problemrestricted to unit disk graphs is NP-hard. A graph is aunit disk graph if its vertices can be mapped to points in the plane such that two vertices are adjacent if and only if the corresponding points have distance at most one. Also, Papadimitriou and Sideri [PS96] conjecture that theMinimum Bisection Problem remains NP-hard when restricted to planar graphs. They studygrid graphs, which are finite induced subgraphs of the infinite grid. Consider the canonical embedding of a grid graphG. Ahole ofGis a face other than the infinite face whose boundary is not a cycle of length four.

Papadimitriou and Sideri show that a minimum bisection in a grid graph onnvertices without holes can be computed inO(n⁵) time and this approach can be generalized to run inO(n^5+2h) time when applied to a grid graph onnvertices withhholes. Furthermore, they show that theMinimum Bisection Problem in planar graphs can be reduced to theMinimum Bisection Problemin grid graphs with an arbitrary number of holes. The algorithm for grid graphs onnvertices without holes has been improved to run inO(n⁴) time by Feldmann and Widmayer [FW15]. Moreover, Bui et al. [Bui+87] showed that, for any fixed integerd≥3, theMinimum Bisection Problem remains NP-hard when restricted tod-regular graphs, i. e., for graphsG= (V, E) where each vertexv∈V satisfies deg(v) =d. This immediately implies that theMinimum Bisection Problemrestricted to graphs with maximum degree 3 is NP-hard.

Another way to deal with the hardness of a problem is to study approximations. Roughly speaking, the idea is to compute a bisection that cuts few edges but might not be a minimum bisection. An algorithm is anα-approximation for theMinimum Bisection Problemif it computes a bisection (B, W) in the input graphGwitheG(B, W)≤αMinBis(G) in polynomial time. Currently, the best known approximation algorithm for the Minimum Bisection Problemis the O(logn)-approximation for arbitrary graphs on n vertices due to Räcke [Räc08]. Nothing better has been established for planar graphs, but an O(logn)-approximation for planar graphs on n vertices had been known before the result of Räcke, see [FK02]. When the minimum degree of the considered graph is linear, apolynomial-time approximation scheme for theMinimum Bisection Problem is known [AKK99], that is, for any fixedε >0, there is a polynomial-time (1 +ε)-approximation for graphsG= (V, E) onnvertices, that satisfy deg(v) = Ω(n)

Consider a graph Gthat allows a bisection of constant width. Then, one can compute a minimum bisection inGin polynomial time by brute-force or, more precisely, by trying all possibilities to remove a constant number of edges fromGand form a set of half of its vertices from the resulting components.

The decision version of theMinimum Bisection Problemis to decide for an input graphGand input parameterewhether the graphGallows a bisection of width at moste. Cygan et al. [Cyg+14] describe an algorithm that solves this question inO

2^O(e3)n³log³n

time when given a graphGonnvertices and an integere, which shows that the decision version of theMinimum Bisection Problemisfixed parameter tractable. Being fixed parameter tractable means that there is an arbitrary functionf such that there is an algorithm for the decision version of theMinimum Bisection Problemthat runs in timeO(f(e)n^O(1)), i. e., polynomial innbut with arbitrary dependence on the parametere, when given a graphGonnvertices and an integer e. Hence, when a graphGon n vertices with MinBis(G) =O(p³ log(n)) is considered, then MinBis(G) can be determined in polynomial time. Note that the brute-force approach results in an algorithm running in Ω

|E| e

time for a graphG= (V, E) and an integere, and does not suffice to show that the decision version of theMinimum Bisection Problemis fixed-parameter tractable.

What upper bounds on the width of a minimum bisection for certain graph classes are known? Any graph G = (V, E) with n := |V| even satisfies MinBis(G) ≤ ¹₂|E|_nⁿ₋₁. Indeed, when a set B ⊆V of size ¹₂nis chosen at random, then each edge inE is cut with probability 2· ⁿ_n² · _n−1ⁿ² = ¹₂ ·_n−1ⁿ , so the expected width of the bisection (B, W) with W :=V \B is ¹₂|E|_n−1ⁿ and, hence, there is a bisection of width at most ¹₂|E|_nⁿ₋₁ inG. The complete graphKn withneven shows that this bound is tight, as every bisection (B, W) in Kn cuts ¹₄n² edges and ¹₂|E(Kn)|_nⁿ₋₁ = ¹₄·n(n−1)·_nⁿ₋₁ =¹₄n². Since every treeT = (V, E) satisfies|E|=|V| −1, this implies that MinBis(T)≤ ¹₂nfor every treeT onnvertices with neven. The starK₁,n−1 onnvertices shows that this bound is tight.

Furthermore, consider a treeT onnvertices with maximum degree ∆0. One can show that owing to the existence of aseparating vertex, i. e., a vertex whose removal leaves no component of size greater than ¹₂n, a bisection of width at most ∆0log₂(n) inT can be constructed and, hence, MinBis(T)≤∆0·log₂(n), see e. g.

Corollary4.9in Chapter4where a slightly different method is used to derive the same bound. It is easy to see that a bisection satisfying this bound can be computed inO(n) time. Furthermore, the bound is tight up to a constant factor, because a perfect ternary treeTh of heighthsatisfies MinBis(Th)≥h−log₃(h), see Theorem 4.11 in [Sch13]. The method can be generalized to planar graphs by using planar separators as in [LT79] to obtain MinBis(G) =O(∆0√

n) for planar graphsGonnvertices with maximum degree ∆0, see also Theorem 6.2 in [Jan+05]. Moreover, the bound for trees can be generalized to tree-like graphs by using tree decompositions to obtain MinBis(G)≤∆0(tw(G) + 1) log₂(n) for every graph Gonnvertices, where tw(G) denotes the tree-width of G. This method will also be used in Section 3.1 to construct bisections, where a proof for the bound for planar graphs and tree-like graphs is presented.

Lower bounds are more difficult to derive than upper bounds and only few are known. One example is the spectral bound MinBis(G)≥ ¹₄λ2n for graphs Gon n vertices with n even, see Proposition 2.1 in [Moh92], whereλ₂ denotes the second smallest eigenvalue of the Laplacian ofG. TheLaplacian of a graphGwithV(G) = [n] for some integernis the matrix obtained from a diagonal matrix, whosei^thentry on the diagonal is degG(i), by subtracting the adjacency matrix ofG. Chapter 1.9.1 in [Lei92] introduces another way to obtain a lower bound on the minimum bisection width in connected graphs. The idea is the following. Consider a connected graphGonnvertices and a bijection of the vertex set ofKn to the vertices ofGas well as a function that maps each edge ofKn to a path joining the corresponding vertices ofG. Thecongestion of an edgeeis defined as the number of such paths that useeand the congestion C of the embedding ofKn intoGis defined to be the maximum congestion among all edges ofG. Then, MinBis(G)≥ _4C¹ n² ifnis even, and MinBis(G)≥_4C¹ (n²−1) ifnis odd.

a)A minimum bisection (B, W), which has width 4.

The vertices inB are colored black and the vertices inW are colored white.

b) A minimum 3-section (B1, B₂, B₃), which has width 2. The vertices are colored black, white, and gray to indicate the setB`to which they belong.

Figure 1.2: Different k-sections of a perfect ternary tree. A perfect ternary tree is an example of a tree for which MinSeck(T) does not increase monotonically askincreases.