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      
 4      do
 5         compute D values for all a in A and b in B
 6         let gv, av, and bv be empty lists
 7         for (n := 1 to |V|/2)
 8            find a from A and b from B, such that g = D[a] + D[b] - 2*E(a, b) is maximal
 9            remove a and b from further consideration in this pass
 10           add g to gv, a to av, and b to bv
 11           update D values for the elements of A = A \ a and B = B \ b
 12        end for
 13        find k which maximizes g_max, the sum of gv[1],...,gv[k]
 14        if (g_max > 0) then
 15           Exchange av[1],av[2],...,av[k] with bv[1],bv[2],...,bv[k]
 16     until (g_max <= 0)
 17  return G(V,E)

References

1. Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal 49: 291–307. doi:10.1002/j.1538-7305.1970.tb01770.x.
2. Ravikumār, Si. Pi; Ravikumar, C.P (1995). Parallel methods for VLSI layout design. Greenwood Publishing Group. p. 73. ISBN 978-0-89391-828-6.