A heuristic solver for the set cover problem
Set Cover Problem ---------- The Set Cover Problem (SCP) is one of the NP-complete problems in computer science. It has many real-life applications, such as crew assigning for trains and airlines, selection of fire station/school locations etc. I adopted it for classification purposes and developed a new Archetype technique (Zhu 2016): https://arxiv.org/abs/1606.07156 Because of its NP-completeness, there is no known efficient exact algorithm for SCP. I have developed a heuristic method and implemented in Python, which I release here. SetCoverPy ---------- SetCoverPy is developed and maintained by Guangtun Ben Zhu, It is designed to find an (near-)optimal solution to the set cover problem (SCP) as fast as possible. It employs an iterative heuristic approximation method, combining the greedy and Lagrangian relaxation algorithms. It also includes a few useful tools for a quick chi-squared fitting given two vectors with measurement errors. For the standard tests (4,5,6, A-H instances from Beasley's OR Library), the code yields solutions that are 96%-100% optimal (see scp_BeasleyOR_test_results.txt). As all the codes, this code can always be improved and any feedback will be greatly appreciated. And here are some small tips using this SCP solver: - If you are not satisfied with the solution, just run it a few more times (when your servers are free) and select the best solution. Or you can also select a larger maxiters at the instantiation: >> g = setcover.SetCover(a_matrix, cost, maxiters=100) - If you are really really lazy and don't want to wait for a near-optimal solution, you can just run the greedy solver, which takes no time: >> g.greedy() How to Install ---------- I recommend using pip to install the code: > pip install SetCoverPy If you are inspired and would like to contribute, you are welcome to clone or fork the repository. Please do drop me a message if you are interested in doing so. Test data ---------- Test data provided with this package only includes one instance from Beasley's OR Library. The rest of the Beasley's OR Library can be retrieved here: https://s3.us-east-2.amazonaws.com/setcoverproblem/BeasleyOR/scp_BeasleyOR_numpyformat.tar.gz The Archetype test data can be retrieved here: https://s3.us-east-2.amazonaws.com/setcoverproblem/ExtragalacticTest/Extragalatic_Archetype_testsample_spec.fits https://s3.us-east-2.amazonaws.com/setcoverproblem/ExtragalacticTest/Extragalatic_Archetype_testsample.fits Run the test ------------- Input: -- a_matrix[mrows, ncols], the binary relationship matrix a_matrix[irow, jcol] = True if jcol covers irow -- cost[ncols], the cost of columns. I recommend using normalized cost: cost/median(cost) Instantiation: >> a_matrix = np.load('./BeasleyOR/scpa4_matrix.npy') >> cost = np.load('./BeasleyOR/scpa4_cost.npy') >> g = setcover.SetCover(a_matrix, cost) Run the solver: >> solution, time_used = g.SolveSCP() ...... Final Best solution: 234 Took 1.287 minutes to reach current solution. (Results of course will depend on the configuration of your machine) Output: -- g.s, the (near-optimal) minimal set of columns, a binary 1D array, g.s[jcol] = True if jcol is in the solution -- g.total_cost, the total cost of the (near-optimal) minimal set of columns Additional tool ------------- The mathutils module includes two function for (quick) estimation of the weighted chi-squared distance, quick_amplitude and quick_totalleastsquares. If you have two vectors x and y with errors x_err and y_err, they perform a least chi-squared fitting for the amplitude a in y = a*x in an iterative manner. For example: > from SetCoverPy import mathutils > a, chi2 = mathutils.quick_amplitude(x, y, xerr, yerr) or > a, chi2 = mathutils.quick_totalleastsquares(x, y, xerr, yerr) The difference between the two functions is the second includes an optimization step with the optimize module in SciPy and is considerably slower, though it provides (slightly) more accurate results.
Python >3.5.1 Numpy >1.11.0 Scipy >0.17.0 I did not fully test the code on earlier versions.
As all the codes, this code can always be improved and any feedback will be greatly appreciated. Sincerely, Guangtun Ben Zhu