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.

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.^[1]^[2]

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, 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 2 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.
1 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.

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