Maximal Clique Problem

Problem Description

This is the problem of finding a maximal clique C in an undirected graph G. That is, for each other clique C' in G, the number of nodes in C should be larger than or equal to the number of nodes in C'.

As input, a directed graph is given, encoded by facts over the predicates node/1 and edge/2, providing respectively the nodes and the edges. A solution to the problem is a maximal clique in the undirected version of the input graph (i.e., in the symmetric closure of the input graph). A solution is encoded by the predicate clique/1: the set of nodes v such that clique(v) holds should form a maximal clique (possibly one of maximum cardinality).

Predicates

• Input: node/1, edge/2

• Output: clique/1

Input format

As input, a directed graph is given. A solution is the largest clique in the symmetric closure of this input graph. The nodes of the input graph are given by facts of the form node(i). The edges of the graph are given by facts of the form edge(i,j).

Output format

The output should consist of a set of facts of the form clique(i). The set of all 'i' that appear in such a fact, should form a maximal clique in the symmetric closure of the input graph.

Example(s)

Input: node(1). node(2). node(3). node(4). node(5). node(6). edge(1,2). edge(1,5). edge(2,3). edge(2,5). edge(3,4). edge(4,5). edge(4,6).

For the above example, the following are two correct outputs:

• clique(1). clique(2). clique(5).

• clique(1). clique(2).

The first is better than the second (in fact, the first is the unique maximum clique).

Notes and Updates

• Specification: updated "Output format" section on date 01/02/2011;

Author(s)

• Author: Johan Wittocx
  • Affiliation: Department of Computer Science, K.U.Leuven, Belgium