# Kernighan–Lin algorithm

This article is about the heuristic algorithm for the graph partitioning problem. For a heuristic for the traveling salesperson problem, see Lin–Kernighan heuristic.

Kernighan–Lin is a O(n2 log(n)) heuristic algorithm for solving the graph partitioning problem. The algorithm has important applications in the layout of digital circuits and components in VLSI.[1][2]

## Description

Let $G(V,E)$ be a graph, and let $V$ be the set of nodes and $E$ the set of edges. The algorithm attempts to find a partition of $V$ into two disjoint subsets $A$ and $B$ of equal size, such that the sum $T$ of the weights of the edges between nodes in $A$ and $B$ is minimized. 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. Furthermore, let

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

be the difference between the external and internal costs of a. 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.[1]

## Pseudocode

See [2]

 1  function Kernighan-Lin(G(V,E)):
2      determine a balanced initial partition of the nodes into sets A and B
3      A1 := A; B1 := B
4      do
5         compute D values for all a in A1 and b in B1
6         for (n := 1 to |V|/2)
7            find a[i] from A1 and b[j] from B1, such that g[n] = D[a[i]] + D[b[j]] - 2*c[a[i]][b[j]] is maximal
8            move a[i] to B1 and b[j] to A1
9            remove a[i] and b[j] from further consideration in this pass
10           update D values for the elements of A1 = A1 \ a[i] and B1 = B1 \ b[j]
11        end for
12        find k which maximizes g_max, the sum of g[1],...,g[k]
13        if (g_max > 0) then
14           Exchange a[1],a[2],...,a[k] with b[1],b[2],...,b[k]
15     until (g_max <= 0)
16  return G(V,E)



## References

1. ^ a b Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal 49: 291–307.
2. ^ a b Ravikumār, Si. Pi; Ravikumar, C.P (1995). Parallel methods for VLSI layout design. Greenwood Publishing Group. p. 73. ISBN 978-0-89391-828-6. OCLC 2009-06-12.