Kernighan–Lin algorithm
# Kernighan–Lin algorithm

The Kernighan–Lin algorithm is a heuristic algorithm for finding partitions of graphs. The algorithm has important practical application in the layout of digital circuits and components in electronic design automation of VLSI.

## 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 $O(n^2$ log n).

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

    $D_{s}=E_{s}-I_{s}$

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

    $T_{old}-T_{new}=D_{a}+D_{b}-2c_{a,b}$

where $c_{a,b}$ 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 $A$ and $B$ which maximizes $T_{old}-T_{new}$ and then executes the operations, producing a partition of the graph to A and B.

## Pseudocode

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