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QUALEX-MS: QUick ALmost EXact maximum weight clique solver based on a generalized Motzkin-Straus formulation, ver 1.2

Copyright (c) Stanislav Busygin, 2000-2009. All rights reserved.


This software is to solve maximum weight clique/independent set problem. It is well-known that this problem is NP-hard, so an exact efficient algorithm for it probably does not exist. However, QUALEX-MS has shown the ability to solve this problem exactly in many cases, including test instances considered hard for all existing algorithms. Complexity of the routine is O(n^3), where n is the number of graph vertices.

The algorithm uses a trust region technique for a generalization of the Motzkin-Straus formulation for maximum clique problem. The generalization allows to consider vertex weights. The RAM requirement is mainly determined by usage of DSYEVR routine of LAPACK for eigendecomposition of an nxn double precision matrix. That is, the available memory should be enough for, at least, two nxn double matrices.

This software is distributed under GNU General Public License, ver. 3.

  1. USAGE

QUALEX-MS uses some linear algebraic routines from the standard packages BLAS and LAPACK. Please install them if you want to build the executable file. They can be gotten at NetLib website:

Unless your hardware platform is very specific, it is suggested to use the so-called ATLAS implementation of BLAS. There are ATLAS prebuilts for almost all hardware platforms available for free and compiled with full possible optimization.

Then, if you use the GNU environment, put correct values for BLASLIB and LAPACKLIB in Makefile and just type make.

To use the solver, issue the command:

qualex-ms [flags] <dimacs_binary_file> [-w<weights_file>]


+c: looking for maximum clique (default)

-c: looking for maximum independent set

+1: vertex numbers in solution file go from 1

-1: vertex numbers in solution file go from 0 (default)

weights_file: a text file for list of vertex weights (reals >= 1.0)

An obtained solution will be stored in a corresponding .sol file. For example, you can find the maximum clique of an instance coded in probe.clq.b file by the command

qualex-ms probe.clq.b

File probe.sol will be created to store the result.

File probe.w contains a sample list of weights for this instance, so the command

qualex-ms probe.clq.b -wprobe.w

will take into accout the given weights.

  1. What is new?

version 1.1:

  • the preliminary greedy heuristic is now MIN starting n times (i.e. from each vertex);
  • the quadratic programming formulation is scaled by square roots of the vertex weights.

version 1.1.1:

  • minor code optimization for the degenerative case.

version 1.1.2:

  • an empty graph bug fixed.

version 1.2:

  • code redesigned to improve readability;
  • bool_vector is now 64-bit compliant;
  • a new parameter allows numbering of vertices from 1 (not 0) in solution files;
  • Windows executable is recompiled with newest MinGW gcc and LAPACK 3.1.1.


QUick ALmost EXact Maximum Weight Clique/Independent Set Solver based on the Motzkin-Straus QP formulation







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