The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

Abstract

Abstract. In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W [1]-hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G) ≤ k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O * (f (k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.

Vertex Cover Kernelization Revisited

“…Related Work The idea of studying parameterized problems using alternative parameters is not new (see, e.g., [36]), but was recently advocated by Fellows et al [22,23,37] in the call to investigate the complexity ecology of parameters. They posed that inputs to computational problems are rarely arbitrary or random because these inputs are created by processes which are themselves computationally bounded.…”

Fixed Parameter Evolutionary Algorithms and Maximum Leaf Spanning Trees: A Matter of Mutation

“…The lemma is proven using an approach similar (but greatly simplified) to the one used in [3]; our focus here is on giving a self-contained presentation sufficient for obtaining the claimed expected runtime. Note also, that kernelization results, such as [3], almost always require a modification of the problem instance while we are interested in bounding the original instance.…”

“…Note also, that kernelization results, such as [3], almost always require a modification of the problem instance while we are interested in bounding the original instance.…”

Tractable Parameterizations for the Minimum Linear Arrangement Problem

“…(Whether the MLA problem can be exactly solved in FPT time for the parameter k alone still remains open.) In [9] it is shown that Bandwidth is FPT, parameterized by the max leaf number of the input graph. Here we obtain a matching result for MLA; it can be exactly solved in FPT time for this parameter.…”