Kernighan–Lin algorithm

Description

The input to the algorithm is an undirected graph G = (V, E) with vertex set V, edge set E, and (optionally) numerical weights on the edges in E. The goal of the algorithm is to partition V into two disjoint subsets A and B of equal (or nearly equal) size, in a way that minimizes the sum T of the weights of the subset of edges that cross from A to B. If the graph is unweighted, then instead the goal is to minimize the number of crossing edges; this is equivalent to assigning weight one to each edge. The algorithm maintains and improves a partition, in each pass using a greedy algorithm to pair up vertices of A with vertices of B, so that moving the paired vertices from one side of the partition to the other will improve the partition. After matching the vertices, it then performs a subset of the pairs chosen to have the best overall effect on the solution quality T. Given a graph with n vertices, each pass of the algorithm runs in time O(n² log n).

In more detail, for each ${\displaystyle a\in A}$, let ${\displaystyle I_a}$ be the internal cost of a, that is, the sum of the costs of edges between a and other nodes in A, and let ${\displaystyle E_a}$ be the external cost of a, that is, the sum of the costs of edges between a and nodes in B. Similarly, define ${\displaystyle I_b}$, ${\displaystyle E_b}$ for each ${\displaystyle b\in B}$. Furthermore, let

${\displaystyle D_s=E_s-I_s}$

be the difference between the external and internal costs of s. If a and b are interchanged, then the reduction in cost is

${\displaystyle T_{old}-T_{new}=D_a+D_b-2c_{a,b}}$

where ${\displaystyle c_{a,b}}$ is the cost of the possible edge between a and b.

The algorithm attempts to find an optimal series of interchange operations between elements of ${\displaystyle A}$ and ${\displaystyle B}$ which maximizes ${\displaystyle T_{old}-T_{new}}$ and then executes the operations, producing a partition of the graph to A and B.