Parallel Maximum Clique (PMC) Library

In short, a parameterized high performance library for computing maximum cliques in large sparse graphs.

Finding maximum cliques, k-cliques, and temporal strong components are in general NP-hard. Yet, these can be computed fast in most social and information networks. The PMC library is designed to be fast for solving these problems. Algorithms in the PMC library are easily adaptable for use with a variety of orderings, heuristic strategies, and bounds.


  1. General framework for parallel maximum clique algorithms
  2. Optimized to be fast for large sparse graphs + Algorithms tested on networks of 1.8 billion edges
  3. Set of fast heuristics shown to give accurate approximations
  4. Algorithms for computing Temporal Strongly Connected Components (TSCC) of large dynamic networks
  5. Parameterized for computing k-cliques as fast as possible
  6. Includes a variety of tight linear time bounds for the maximum clique problem
  7. Ordering of vertices for each algorithm can be selected at runtime
  8. Dynamically reduces the graph representation periodically as vertices are pruned or searched + Lowers memory-requirements for massive graphs, increases speed, and has caching benefits



First, you’ll need to compile the parallel maximum clique library.

	$ cd path/to/pmc/
	$ make

Afterwards, the following should work:

	# compute maximum clique using the full algorithm `-a 0`
	./pmc -f data/socfb-Texas84.mtx -a 0

PMC has been tested on Ubuntu linux (10.10 tested) and Mac OSX (Lion tested) with gcc-mp-4.7 and gcc-mp-4.5.4

If you run into problems compiling, ensure openMP is installed. In OSX, this can be done as follows:

sudo port selfupdate

Now, install GCC via the terminal command:

sudo port install gcc47

Now type "port select --list gcc", you should see mp-gcc47, among others. To select MP-GCC 4.7, run the command:

sudo port select --set gcc mp-gcc47

And you're all set. Note that other gcc versions also work (e.g., mp-gcc45, mp-gcc46,…).

Please let me know if you run into any issues.

Input file format


The parallel maximum clique algorithms use tight bounds that are fast to compute. A few of those are listed below.

All bounds are dynamically updated.

Examples of the three main maximum clique algorithms are given below. Each essentially builds on the other.

# uses the four basic k-core pruning steps
./pmc -f ../pmc/data/output/socfb-Stanford3.mtx -a 2

# k-core pruning and greedy coloring
./pmc -f ../pmc/data/output/socfb-Stanford3.mtx -a 1

# neighborhood core pruning (and ordering for greedy coloring)
./pmc -f ../pmc/data/output/socfb-Stanford3.mtx -a 0

Dynamic graph reduction

The reduction wait parameter -r below is set to be 1 second (default = 4 seconds).

./pmc -f data/sanr200-0-9.mtx -a 0 -t 2 -r 1

In some cases, it may make sense to turn off the explicit graph reduction. This is done by setting the reduction wait time ‘-r’ to be very large.

# Set the reduction wait parameter
./pmc -f data/socfb-Stanford3.mtx -a 0 -t 2 -r 999


The PMC algorithms are easily adapted to use various ordering strategies. To prescribe a vertex ordering, use the -o option with one of the following:

Direction of ordering

Vertices are searched by default in increasing order, to search vertices in decreasing order, use the d option:

./pmc -f data/p-hat700-2.mtx -a 0 -d


The fast heuristic may also be customized to use various greedy selection strategies. This is done by using -h with one of the following:

Terminate after applying the heuristic

Approximate the maximum clique using ONLY the heuristic by not setting the exact algorithm via the -a [num] option. For example:

./pmc -f data/sanr200-0-9.mtx -h deg

Turning the heuristic off

# heuristic is turned off by setting `-h 0`.
./pmc -f data/tscc_enron-only.mtx -h 0 -a 0


The parallel maximum clique algorithms have also been parameterized to find cliques of size k. This routine is useful for many tasks in network analysis such as mining graphs and community detection.

# Computes a clique of size 50 from the Stanford facebook network
./pmc -f data/socfb-Stanford3.mtx -a 0 -k 50

using -o rand to find potentially different cliques of a certain size

# Computes a clique of size 36 from sanr200-0-9
./pmc -f data/sanr200-0-9.mtx -a 0 -k 36 -o rand

Terms and conditions

Please feel free to use these codes. We only ask that you cite:

Ryan A. Rossi, David F. Gleich, Assefaw H. Gebremedhin, Md. Mostofa Patwary,  
A Fast Parallel Maximum Clique Algorithm for Large Sparse Graphs and Temporal  
Strong Components, arXiv preprint 1302.6256, 2013.  

These codes are research prototypes and may not work for you. No promises. But do email if you run into problems.

Copyright 2011-2013, Ryan A. Rossi. All rights reserved.