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 formula_1, let formula_2 be the "internal cost" of "a", that is, the sum of the costs of edges between "a" and other nodes in "A", and let formula_3 be the "external cost" of "a", that is, the sum of the costs of edges between "a" and nodes in "B". Similarly, define formula_4, formula_5 for each formula_6. Furthermore, let 
formula_7
be the difference between the external and internal costs of "s". If "a" and "b" are interchanged, then the reduction in cost is
formula_8
where formula_9 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 formula_10 and formula_11 which maximizes formula_12 and then executes the operations, producing a partition of the graph to "A" and "B".
Pseudocode.
Source:
 function Kernighan-Lin("G(V, E)") is
 determine a balanced initial partition of the nodes into sets A and B
 do
 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[a] + D[b] − 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
 end for
 find k which maximizes g_max, the sum of gv[1], ..., gv[k]
 if g_max ; 0 then
 Exchange av[1], av[2], ..., av[k] with bv[1], bv[2], ..., bv[k]
 until (g_max ≤ 0)
 return G(V, E)
References.
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