An implementation of a branch and bound algorithm for the traveling salesperson problem as described by Little, Murty, Sweeney, and Karel in their 1963 paper
Given a graph with weights between edges integers, this program finds the shortest Hamiltonian cycle starting and ending at vertex 0.
A pdf of the original paper can be found in this repository (source.pdf) as well as online here or here. This website gives many formats in which the source may be downloaded. When implementing the algorithm, I found this video helped me understand it better.
The implementation is given in C++. It stays relatively true to the description that the authors give for how to implement it to minimize computation and space complexity.
Dependencies:
- cmake. Should be available in your package manager on Linux.
- gflags. Available as
gflagsfor Debian-based distributions, otherwise, clone the repo (instructions given in the URL), create abuild/subdirectory, runcmake .. && sudo make installto install on your system. - (Optional) Google Test (for unit testing). Cloning repo and modifying the
GMOCK_ROOTvariable in the baseCMakeLists.txtfile should be sufficient for CMake to find it.
After cloning the repository into a local directory, cd into that directory and run ./configure.sh, then cd into build/ and run make. To run individual cases of the program manually, run ./build/src/littletsp < input.txt from the top level of the directory where input.txt is the path to a valid input file as described below.
This program also provides a brute force algorithm for solving the TSP that conducts a simple depth first search and returns the minimum path (provide --solver=naive on the command line). A heuristic-based approach is available too (--solver=fast). Also, typing littletsp --help will print a help message displaying options for running.
Finally, running the demo script (python3 test/demo.py) will provide a demonstration of Little's algorithm's speedup vs the naive implementation. See options for this script by typing python3 test/demo.py --help.
To build the unit tests, download Google Mock anywhere on your system. Then, open CMakeLists.txt and change the GMOCK_ROOT variable to the location where you downloaded it on your system. Then, build the unittest binary by running make unittest from build/ and run the unittest binary with ./src/unittest
The input file format is as follows. Line 1 contains the size N of the map. The map is considered an NxN square. The next line contains the number M of vertices. The next M lines contain the coordinate of each vertex i in the format x y where x is the x-coordinate of i, and y is the y-coordinate of i. There may be any number of empty lines at the end of the file. For example:
100
4
1 29
2 82
12 4
3 76
is a valid input file. The vertex with coordinates with coordinates 1 29 would be considered vertex 0. Input must be redirected to standard input std::cin. See the "Instructions to configure and run" for an example of how to run.
The graph is interpreted as a complete graph. That is, it is a simple undirected graph in which every pair of vertices is connected by a unique edge. The weight of each edge is the Manhattan (taxi cab) distance between the two vertices it connects.
The shortest path found printed to standard output (std::cout) in the following format. Line 1 contains the total length of the cycle found. Line 2 contains each vertex i in the order that they must be visited in order to find the shortest cycle. This output always starts at vertex 0, and is assumed to finish at vertex 0 (though the final 0 is not printed). For example:
178
0 1 3 2
would be the output for the input file given above.
All source code can be found in src/.
Top level modules (main, path, util) contain code useful for running the entire binary.
The src/graph/ subdirectory contains an object-oriented implementation of a Graph. See graph/graph for the interface, and graph/manhattan for an implementation of this interface. graph/factory is used as a factory for constructing graph objects.
The src/tsp_solver/ subdirectory contains an object-oriented implementation of various solvers for the TSP problem. src/tsp_solver/naive gives the naive implementation, and src/tsp_solver/fast the fast heuristic-based approach. src/tsp_solver/tsp_solver gives the interface solvers must implement, and src/tsp_solver/factory provides a factory for constructing various solvers. The implementation of Little's algorithm can be found in src/tsp_solver/little/ (it had enough source files to deserve its own implementation).
In src/tsp_solver/little/solver LittleTSPSolver provides the top-level operations for solving the TSP with Little's algorithm. It creates TreeNodes, which then create CostMatrixs, which are used to find the next edge to branch on. LittleTSPSolver creates these branches (at most 2, an branch including that edge, and a branch exluding that edge), and branches toward the include branch. When a valid solution is found, it is stored and used to prune branches which have a higher lower bound than the cost of the solution.
little-tsp, an implementation of the branch and bound algorithm for the TSP as described in a 1963 paper by Little et al Copyright (C) 2012-2016 Kar Epker
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.