Not So Easy Problems for Tree Decomposable Graphs

We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some recent results for problems regarding list and equitable colorings, general factors, and generalized satisfiability. Second we establish a new hardness result for the problem of minimizing the maximum weighted outdegree for orientations of edge-weighted graphs of bounded treewidth.

💡 Research Summary

The paper investigates a class of combinatorial problems that are polynomial‑time solvable on graphs of bounded treewidth, yet whose polynomial degree necessarily depends on the treewidth bound. After a brief introduction to treewidth and its role in algorithmic graph theory, the authors adopt the framework of parameterized complexity to distinguish problems that are fixed‑parameter tractable (FPT) from those that are W