The Kernighan–Lin algorithm is a heuristic algorithm
for finding partitions of graphs
The algorithm has important applications in the layout of digital circuits and components in VLSI
The input to the algorithm is an undirected graph
with vertex set , edge set , and (optionally) numerical weights on the edges in . The goal of the algorithm is to partition into two disjoint subsets and of equal (or nearly equal) size, in a way that minimizes the sum of the weights of the subset of edges that cross from to . 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 with vertices of , 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 .
Given a graph with vertices, each pass of the algorithm runs in time .
In more detail, for each
be the ''internal cost'' of ''a'', that is, the sum of the costs of edges between ''a'' and other nodes in ''A'', and let
be the ''external cost'' of ''a'', that is, the sum of the costs of edges between ''a'' and nodes in ''B''. Similarly, define
. Furthermore, let
be the difference between the external and internal costs of ''s''. If ''a'' and ''b'' are interchanged, then the reduction in cost is
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
and then executes the operations, producing a partition of the graph to ''A'' and ''B''.
function Kernighan-Lin(''G(V, E)'') is
determine a balanced initial partition of the nodes into sets A and B
compute D values for all a in A and b in B
let gv, av, and bv be empty lists
for n := 1 to |V| / 2 do
find a from A and b from B, such that g = D
− 2×c(a, b) is maximal
remove a and b from further consideration in this pass
add g to gv, a to av, and b to bv
update D values for the elements of A = A \ a and B = B \ b
find k which maximizes g_max, the sum of gv
if g_max > 0 then
until (g_max ≤ 0)
return G(V, E)
* Fiduccia–Mattheyses algorithm