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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(n2 log n).

In more detail, for each a ∈ A {\displaystyle a\in A} , let I a {\displaystyle 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 {\displaystyle 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 {\displaystyle I_{b}} , E b {\displaystyle E_{b}} for each b ∈ B {\displaystyle b\in B} . Furthermore, let

D s = E s − I s {\displaystyle 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 o l d − T n e w = D a + D b − 2 c a , b {\displaystyle T_{old}-T_{new}=D_{a}+D_{b}-2c_{a,b}}

where c a , b {\displaystyle 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 {\displaystyle A} and B {\displaystyle B} which maximizes T o l d − T n e w {\displaystyle T_{old}-T_{new}} and then executes the operations, producing a partition of the graph to A and B.³

Pseudocode

Source:⁴

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)

See also

• Fiduccia–Mattheyses algorithm

References

1. Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal. 49 (2): 291–307. doi:10.1002/j.1538-7305.1970.tb01770.x. /wiki/Brian_Kernighan ↩

2. Ravikumar, C. P (1995). Parallel methods for VLSI layout design. Greenwood Publishing Group. p. 73. ISBN 978-0-89391-828-6. 978-0-89391-828-6 ↩

3. Kernighan, B. W.; Lin, Shen (1970). "An efficient heuristic procedure for partitioning graphs". Bell System Technical Journal. 49 (2): 291–307. doi:10.1002/j.1538-7305.1970.tb01770.x. /wiki/Brian_Kernighan ↩

4. Ravikumar, C. P (1995). Parallel methods for VLSI layout design. Greenwood Publishing Group. p. 73. ISBN 978-0-89391-828-6. 978-0-89391-828-6 ↩