Crossing minimization in layered graphs

Problem Description

The standard approach for drawing hierarchical network diagrams is a three phase approach in which

1. nodes in the graph are assigned levels producing a k-level graph;

2. nodes are assigned an order so as to minimize edge crossings in the k-level graph; and

3. the edge routes and node positions are computed.

There has been considerable research into step 2 which is called k-level crossing minimization. Unfortunately this step is NP-hard for even two layers (k = 2) where the ordering on one layer is given. Thus, research has focussed on developing heuristics to solve it. In practice the approach is to iterate through the levels, re-ordering the nodes on each level using heuristic techniques such as the barycentric method. Recently it was shown that modern optimization technology can tackle the complete k-level crossing minimization problem at least for small to medium sized graphs, generating optimal crossings.

A crossing in a graph occurs if there are:

1. nodes N1, N2 on layer i, and

2. nodes N3, N4 on layer j, (j <> i), and

3. edges e1 = (N1,N3), e2 = (N2,N4), and
4. the position of N1 is antecedent of the position of N2, and
5. the position of N4 is antecedent of the position of N3.

Predicates

• Input: layers/1, width/2, in_layer/2, edge/2

• Output: position/2

Input format

The predicates layers/1 and in_layer/2 give the partitioning of nodes into layers. For convenience, width/2 gives the number of nodes in a layer. For example

layers(2). width(0, 2). in_layer(0,n1). in_layer(0,n2). width(1, 1). in_layer(1,n3).

defines two layers, layer 0 containing nodes {n1, n2} and layer 1 containing {n3}.

The predicate edge/2 defines edges between nodes. For example

edge(n1,n3). edge(n2,n3).

Output format

A solution to the problem is an assignment of each node to a position in its layer, which minimizes the number of crossings in the graph. This is given by the position/2 predicate, where position(n,3) assigns node 'n' to the third position in its layer. Positions at a layer L are numbered from 1 until the width of L.

Example(s)

Input:

layers(2). width(0, 2). in_layer(0,n1). in_layer(0,n2). width(1, 1). in_layer(1,n3). edge(n1,n3). edge(n2,n3).

Possible output:

position(n1,1). position(n2,2). position(n3,1).

Additional sample instances: download

Problem Peculiarities

Type: Optimization Competition: Both

This problem is considered quite hard in practice, and is usually approached using ad-hoc heuristics. The instance family is based on graphs taken from the graphviz website and randomly generated layered graphs of known optimal value. The outcomes of this benchmark will give a clear idea of the gap-width of declarative technologies in this respect.

Notes and updates

Author(s)

• Author: Carmine Dodaro
  • Affiliation: University of Calabria, Italy
• Original Authors: Peter Stuckey, Graeme Gange
  • Affiliation: University of Melbourne, Australia