Tree Decomposition: A Feasibility Study

Graphs of small treewidth resemble a tree in a certain (precise) sense. Many computationally hard graph problems can be solved efficiently on graphs of small treewidth using a “tree decomposition,” which represents the constructive aspect of treewidth. However, computing minimum-width tree decompositions is NP-hard in general, but for fixed treewidth, there exist algorithms with polynomial running time. In order to evaluate the practical usability of tree-decomposition algorithms for graphs of arbitrary treewidth, we have implemented several fundamental algorithms related to computing tree decompositions. We present the theory behind solving graph problems using tree decompositions and show how it can be applied in practice to compute “path decompositions.” Then we give a survey of the tree-decomposition algorithms considered and discuss their practical value on the basis of benchmarks. Test graphs were produced using a suite of graph-generating programs that we developed as part of this thesis. Our experiments indicate that the algorithms for graphs of unrestricted treewidth are not viable for input graphs with treewidth beyond the scope of present special-purpose algorithms, which exist for treewidth up to four.